0000223482 00000 n Q 0.562 0.087 TD 0.015 w q >> /F1 0.217 Tf 0 g Q 0.165 0.129 m Q q /Type /XObject 45.324 0 0 45.147 54.202 460.721 cm Q >> q S Q W* n >> Q stream /Font << Q q /F1 0.217 Tf /FormType 1 /Font << endobj >> endobj /Length 55 q BT 0.314 0 l 0.458 0 0 RG >> /Subtype /Form Q 221 0 obj << 45.249 0 0 45.527 217.562 491.586 cm /Meta824 Do Solve using the quadratic formula:
a) EMBED Equation.3 =EMBED Equation.3
EMBED Equation.3=EMBED Equation.3 ; a = ____, b = ____, c = ____
EMBED Equation.3
EMBED Equation.3
Worksheet 35 (7.4)
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3 The solution set is ____________. 0 G /Length 562 q /Meta460 475 0 R stream /Length 34064 /BBox [0 0 0.413 0.283] Q Q q Q 0.458 0 0 RG Q /Meta89 Do /Length 76 W* n >> Q 1 g Q 0 w >> >> /Length 76 /Matrix [1 0 0 1 0 0] 45.249 0 0 45.527 329.731 468.249 cm 0.031 0.437 TD 0000188539 00000 n q 1070 0 obj << 0 0 l /Matrix [1 0 0 1 0 0] /Meta74 Do /I0 Do 0 w /Font << W* n q 0.015 w q /Subtype /Form >> Q /Font << 0000227897 00000 n stream /FormType 1 /Type /XObject Q Q endstream q q /FormType 1 /Matrix [1 0 0 1 0 0] q Q 1 J stream >> 0.267 0.283 l 0.566 0.366 l /Resources << Q endstream q 0 G 0 0 l W* n stream /Length 55 /Matrix [1 0 0 1 0 0] /F1 6 0 R 578.159 506.642 l Q /Font << /BBox [0 0 9.523 0.633] q q 0 G >> 0000176178 00000 n endobj 0 0 l 0 g 278 0 obj << [(-)] TJ 0 g /Length 55 >> /Subtype /Form q 916 0 obj << /I0 Do 3 1. q 45.249 0 0 45.147 329.731 86.573 cm q endstream 0.015 w stream endstream 0 0 l Q stream /Matrix [1 0 0 1 0 0] 0 G [(B\))] TJ /Subtype /Form W* n ET 0 0 l q q 45.663 0 0 45.168 426.844 362.102 cm 0000064394 00000 n BT /Meta128 Do Q 45.249 0 0 45.131 217.562 216.057 cm /Type /XObject /Matrix [1 0 0 1 0 0] 0.458 0 0 RG 0 0.283 m /Matrix [1 0 0 1 0 0] /Meta1051 Do /Meta1038 1055 0 R /Matrix [1 0 0 1 0 0] 45.249 0 0 45.147 329.731 107.652 cm q /FormType 1 0 0 l 0.001 Tc /Resources << 0 0.283 m Q ET /FormType 1 /Resources << /Resources << /Meta8 Do /Meta836 851 0 R >> /Length 61 0 g /Font << 0000052488 00000 n BT q Q /Matrix [1 0 0 1 0 0] ET endobj >> 0 0 l endstream /Length 55 ET q /F1 6 0 R q /Font << 0 0.087 TD 45.249 0 0 45.413 329.731 263.484 cm /Type /XObject 0 G /Resources << stream /Type /XObject Q stream q 0000055059 00000 n 0000196274 00000 n 0.031 0.087 TD We took this picture on the internet we think would be probably the most representative pics for Dividing Complex Numbers Worksheet. /Subtype /Form Q 0.267 0 l Q /Meta1073 Do >> 0.267 0 l Just in case you forgot how to determine the conjugate of a given complex number, see the table … Dividing Complex Numbers Read More » 0000168862 00000 n /F1 0.217 Tf Q 0.015 w 0 g q 0000043878 00000 n /Subtype /Form stream /Font << /Matrix [1 0 0 1 0 0] q >> BT stream /Font << q BT /Length 94 0 g /Subtype /Form /FormType 1 0.748 0.308 TD Q 0.458 0 0 RG endobj /Subtype /Form /Length 102 /Meta174 185 0 R q >> 0 G /Matrix [1 0 0 1 0 0] /Font << 0.001 Tc Q /Font << 1056 0 obj << /FormType 1 /Length 55 /Type /XObject Find the dimensions of the plot of ground if the area including the sidewalk is 819 square meters. endobj /FormType 1 /FormType 1 /Matrix [1 0 0 1 0 0] 0.267 0 l /Meta718 733 0 R stream /BBox [0 0 1.547 0.33] 1.547 0.633 l /F3 0.217 Tf q /Meta842 857 0 R /FormType 1 BT endstream 45.663 0 0 45.168 202.506 362.102 cm /Matrix [1 0 0 1 0 0] /F1 0.217 Tf Q 0000100512 00000 n endobj endobj /BBox [0 0 9.523 0.633] /Meta328 341 0 R 1.547 0 l 1 g q 550 0 obj << q 9.791 0.283 l S q /Matrix [1 0 0 1 0 0] 9.523 0 l 0.564 G /Length 102 /FormType 1 /Subtype /Form 0 0.283 m 941 0 obj << 0.381 0.087 TD /BBox [0 0 1.547 0.33] /Meta975 990 0 R Q 0 G 45.214 0 0 45.147 81.303 691.834 cm 0000168630 00000 n >> /FormType 1 0 0 l q /Length 63 >> /FormType 1 W* n 45.527 0 0 45.147 523.957 254.45 cm 11.988 0.283 l /F1 0.217 Tf /Meta546 Do /Length 8 BT >> /Type /XObject /BBox [0 0 1.547 0.283] 0000057576 00000 n Q /Subtype /Form In Exercises 67-8, divide and simplify into the form a + bi. /Type /XObject /F1 0.217 Tf 0 g /Length 102 stream 877 0 obj << Q stream 781 0 obj << 0000079638 00000 n /FormType 1 q /Subtype /Form /Resources << -0.005 Tc q 349 0 obj << /Length 53 /F1 6 0 R Q Q 0 g /Matrix [1 0 0 1 0 0] /Length 53 /Meta154 165 0 R /Meta560 575 0 R 0000259541 00000 n >> Q /Meta640 Do /Meta110 121 0 R /Meta495 510 0 R q 0.232 0.308 TD 286 0 obj << /Font << 0.564 G 0.267 0.283 l 45.249 0 0 45.413 105.393 513.418 cm Q /Meta502 Do 1 j W* n 0.015 w 45.249 0 0 45.527 217.562 535.249 cm >> Q /Meta475 Do 0 g /Meta874 889 0 R Q Q /Resources << 0 g >> 0 G 0 0.283 m /FormType 1 q /Meta513 Do 0 G /Subtype /Form 0 G Q endobj /F1 0.217 Tf >> q 0.458 0 0 RG /Subtype /Form 0.564 G q endstream /Type /XObject q >> 45.214 0 0 45.131 81.303 317.686 cm 45.663 0 0 45.147 426.844 203.259 cm 0 g >> /Resources << stream /Meta564 Do endstream 45.663 0 0 45.168 202.506 216.057 cm /Length 64 Q W* n /Meta1049 Do /Meta397 Do >> /Font << 330 0 obj << Q /F1 6 0 R /Font << /Resources << 0 w >> /FormType 1 /BBox [0 0 1.547 0.283] /Meta116 Do 45.249 0 0 45.413 441.9 263.484 cm >> >> >> 0 0.283 m 9.791 0 l q q Q Q stream 9.791 0 l /Length 67 W* n /F1 6 0 R [(i)] TJ 0 G /Meta278 Do >> stream /Matrix [1 0 0 1 0 0] 0000268293 00000 n >> endstream (3 + i) - (9 + 4i)
Worksheet 38 (7.1)
Summary 3:
Multiplying complex numbers:
1. Q 538.26 528.474 m endstream stream 0000091531 00000 n stream /Meta162 173 0 R /Length 102 0000065556 00000 n Q q /BBox [0 0 9.523 0.33] 1.547 0.33 l Q Q /Font << Q 0 0 l q /Meta951 Do >> /Meta1060 1077 0 R Q /BBox [0 0 11.988 0.283] [( 2)] TJ 0 g endstream Q /Length 102 45.249 0 0 45.131 217.562 289.079 cm 0 g /Matrix [1 0 0 1 0 0] ET 0 0.087 TD 0 G q /Resources << 0000066019 00000 n ET Q /F3 21 0 R /Type /XObject stream /Matrix [1 0 0 1 0 0] 1 J q 45.213 0 0 45.147 36.134 746.037 cm /Meta234 Do ET 0 g -0.008 Tc /FormType 1 q 0 g /Matrix [1 0 0 1 0 0] /FormType 1 >> /Meta1061 Do /Matrix [1 0 0 1 0 0] /Length 8 BT >> Q /Font << /Length 55 0 0 l q /Length 102 45.249 0 0 45.413 217.562 558.586 cm /Resources << Q 0 0.283 m /Font << /Matrix [1 0 0 1 0 0] 45.214 0 0 45.147 81.303 550.305 cm /F1 0.217 Tf >> 0000348859 00000 n /Meta295 308 0 R 0.458 0 0 RG /FormType 1 0.458 0 0 RG q [( 8)] TJ /Font << q 45.214 0 0 45.147 81.303 550.305 cm /Widths [ 493]>> W* n 0.015 w 1.547 0.633 l ET /FormType 1 /Meta610 Do endstream q 0 0.5 m 1.547 0.283 l 0.283 0.087 TD >> W* n endstream /F1 0.217 Tf 0 g BT 45.249 0 0 45.147 105.393 679.036 cm endobj Q W* n Q -0.002 Tc /FormType 1 0000092412 00000 n 45.663 0 0 45.147 314.675 263.484 cm 284 0 obj << /Length 65 endstream 1 g /Font << /Type /XObject /FormType 1 45.249 0 0 45.131 329.731 143.034 cm /Meta287 300 0 R endobj q /Type /XObject /Meta633 Do 0 G q >> 0 w /Font << >> endstream /Subtype /Form q 0.458 0 0 RG ET /Meta713 728 0 R endstream /Meta969 Do 863 0 obj << /Matrix [1 0 0 1 0 0] 0 0.366 m /BBox [0 0 0.263 0.283] 0 g /Meta1067 1084 0 R q /Type /XObject 813 0 obj << 0 G /F3 21 0 R endstream /Length 303 /Meta276 Do ET /Type /XObject /Meta822 837 0 R /Matrix [1 0 0 1 0 0] Q /Matrix [1 0 0 1 0 0] 45.249 0 0 45.147 441.9 107.652 cm 0 G Q 0000181812 00000 n q /BBox [0 0 1.547 0.33] Q /I0 36 0 R /Length 67 /Resources << /BBox [0 0 1.547 0.283] 0.031 0.087 TD /Matrix [1 0 0 1 0 0] >> Q stream Write an appropriate solution set. /FormType 1 /Resources << Example - ˇˆ˙ ˝˛˚˙ = ˇˆ˙ ˝˛˚˙ ∙ ˝ˇ˚˙ ˝ˇ˚˙ (Multiply by complex conjugate) = ˇ˝˜˙ˇˆ˙ˇ˝˚˙ ˝ˇ˚˙˛˚˙˛˚ ˙ = S Q 0000004657 00000 n /F3 0.217 Tf 0 0.283 m 11.988 0 l /Meta649 Do endobj -0.007 Tc 335 0 obj << q 45.663 0 0 45.147 314.675 622.575 cm W* n >> q BT endstream endobj q q 1.547 0.283 l >> 0 g Q endobj endstream stream >> 45.214 0 0 45.147 81.303 550.305 cm 45.249 0 0 45.527 217.562 491.586 cm 0.267 0 l 0 g Q # � : � z � � � � � 0000357105 00000 n /Meta952 Do Q 0.267 0.283 l endobj endstream 1 j /BBox [0 0 1.547 0.283] 0000050673 00000 n /Meta818 833 0 R /FormType 1 1 g q q q ET Q /Length 562 /Meta1064 Do Q q [(i\))] TJ 45.663 0 0 45.147 202.506 298.866 cm /Matrix [1 0 0 1 0 0] /Meta870 885 0 R q q /FormType 1 0.015 w /Subtype /Form q endobj Click here to buy the accompanying White Rose Maths workbook. q 0 0.633 m /Meta175 Do /Font << q /Type /XObject endobj q >> /BBox [0 0 9.787 0.283] endobj 976 0 obj << 0 0.283 m q 445 0 obj << W* n endstream /Meta278 289 0 R /FormType 1 /FormType 1 0 0.283 m /Subtype /Form /Length 55 >> Adding in standard form. q /Meta60 Do 0 0.283 m /Type /XObject /Meta472 Do Q 0.381 0.087 TD 0000257752 00000 n >> 0 w /Meta262 273 0 R /Type /XObject /BBox [0 0 1.547 0.633] endstream BT ET 0.564 G -0.008 Tc stream 0 g q 0.5 0.087 TD 0 G 0.381 0.087 TD Q /FormType 1 45.249 0 0 45.413 105.393 263.484 cm /Subtype /Form 1 g /Length 66 Q 45.249 0 0 45.131 217.562 289.079 cm /Meta503 518 0 R /BBox [0 0 0.263 0.283] /Length 55 1112 0 obj << /Type /XObject ET Q /MediaBox [0 0 614.294 794.969] /Type /XObject 0.531 0.158 TD BT 0.232 0.308 TD /FormType 1 6.157 0.087 TD /F1 6 0 R q BT >> 45.413 0 0 45.147 523.957 629.351 cm stream Q /Meta241 252 0 R /FormType 1 /FormType 1 /BBox [0 0 1.547 0.33] 0.267 0.5 l q Q 11.988 0.283 l stream Q 0.417 0 l /Subtype /Form 0 G Q 0 g 45.249 0 0 45.131 217.562 216.057 cm 0000244579 00000 n /Resources << [(B\))] TJ 1 g 0 G /Meta109 120 0 R >> q >> 0.001 Tc stream q Q 0.681 0.366 l 45.663 0 0 45.147 314.675 718.183 cm 0.216 0.165 l 45.249 0 0 45.131 441.9 143.034 cm >> Q >> /FormType 1 0 g /BBox [0 0 11.988 0.283] /Type /XObject 0 0.087 TD q 45.249 0 0 45.527 217.562 535.249 cm /Subtype /Form EMBED Equation.3
3. /Subtype /Form endobj endobj q 0 G ET 0.564 G 0 0.283 m >> /Subtype /Form 0 0.283 m 0.531 0.283 l 0.712 0.087 TD /Type /XObject Q q BT 0 w ET Q /F3 21 0 R endstream /Type /XObject 0.015 w q /Length 136 BT /F1 0.217 Tf 0 0 l /Meta646 Do q endobj 0.015 w q /F3 21 0 R ET 241 0 obj << >> q /Meta1049 1066 0 R /FormType 1 [(-)] TJ endstream /MissingWidth 252 1.547 -0.003 l [( \()] TJ Q 1 g Q 0 0.283 m W* n /FormType 1 45.249 0 0 45.147 441.9 718.183 cm endobj ET 0000356616 00000 n 0 G Q 0 w ET 0 0.283 m /F1 6 0 R /Meta155 Do 0 g 0.267 0.5 l 0 0.283 m q 0 w 1 g q Q 1 J /BBox [0 0 0.531 0.283] stream /Meta710 725 0 R Q Q endstream BT 563 0 obj << Q Q /Meta1058 Do 45.249 0 0 45.527 329.731 491.586 cm /Meta282 Do q 1.547 0.33 l /Meta1048 1065 0 R 0 w 0 G Q 0 G /BBox [0 0 9.523 0.283] /FormType 1 0000177881 00000 n /Resources << Q q 0000147904 00000 n /Font << 0 0.283 m 0 g /Matrix [1 0 0 1 0 0] >> stream /Meta377 390 0 R Factor all numerators and denominators completely. 623 0 obj << /FormType 1 0.031 0.087 TD 698 0 obj << /FormType 1 >> /Type /XObject 45.249 0 0 45.147 105.393 107.652 cm 1 g /Type /XObject /Meta554 569 0 R 0 G BT 0.458 0 0 RG q /BBox [0 0 11.988 0.283] /F1 6 0 R /Meta53 Do /Length 66 Q >> /Type /XObject q /Matrix [1 0 0 1 0 0] stream stream 542.777 643.654 m q >> 45.213 0 0 45.211 36.134 676.778 cm endstream /Meta855 870 0 R /F4 0.217 Tf /FormType 1 ET q 0 -0.003 l /Type /XObject 0 0.366 m 1.547 0 l 0000163866 00000 n endobj endstream /I0 Do >> /Meta80 Do 1 J /Resources << 45.663 0 0 45.147 314.675 368.125 cm 0.2 0.437 TD /FormType 1 Multiply a whole number and a decimal - easy (one decimal digit) Multiply a whole number and a decimal - harder (one decimal digit) Multiply a whole number and a decimal - missing factor (one decimal digit) Multiply a whole number and a decimal (1-2 decimal digits) Multiply a whole number and a decimal - missing factor (1-2 decimal digits) /Length 408 /Meta335 Do 0000016749 00000 n q /Matrix [1 0 0 1 0 0] /Type /XObject 0 G 0 G /F1 6 0 R /Meta823 838 0 R /F1 0.217 Tf stream 45.413 0 0 45.147 523.957 289.079 cm >> /Resources << /BBox [0 0 0.263 0.283] endobj /Resources << 0.002 Tw Q /Meta207 Do /BBox [0 0 0.263 0.283] Q q /BBox [0 0 9.523 0.283] W* n /Matrix [1 0 0 1 0 0] BT /Matrix [1 0 0 1 0 0] Q 45.249 0 0 45.147 217.562 203.259 cm q /BBox [0 0 0.263 0.283] /Meta169 Do /FormType 1 45.249 0 0 45.131 105.393 216.057 cm Q stream W* n /Font << ET 1 J /Type /XObject /Length 55 /Subtype /Form /Subtype /Form Q stream /BBox [0 0 9.523 0.283] endobj /BBox [0 0 11.988 0.283] /Resources << 0.531 0 l /Length 55 Q Q Q /Subtype /Form [(A\))] TJ /Matrix [1 0 0 1 0 0] >> 0 0.283 m Q stream q Q stream stream 0000019483 00000 n /Meta817 Do Q 0.031 0.437 TD /Subtype /Form /Length 55 1 g q 0 g 0000020210 00000 n 0000057043 00000 n endstream q 45.663 0 0 45.168 314.675 362.102 cm q /Length 55 stream 0 g 0 0.283 m q 0000165969 00000 n 0 -0.003 l /Meta607 Do q endobj 1 g [(\()] TJ endobj /Meta279 Do Q >> >> /Resources << /Type /XObject Q /BBox [0 0 0.531 0.283] endobj 1.547 0.283 l ET q >> /Length 55 /FormType 1 /Font << 0.417 0.283 l >> BT 0.564 G q /Subtype /Form stream 636 0 obj << /Meta695 710 0 R q 0 G 0.458 0 0 RG /FormType 1 -0.002 Tc Q Q Q 0 g stream W* n >> Q >> 0 w q /F1 0.217 Tf /F1 0.217 Tf endstream 0 g /Length 55 Q endobj /Meta211 Do Q /F1 6 0 R q q [( 24)] TJ /Subtype /Form /Meta757 Do 1.547 0 l 45.249 0 0 45.131 217.562 143.034 cm endobj /FormType 1 971 0 obj << /Resources << /Font << stream q 1 J q /FormType 1 0 g 0 g /FormType 1 W* n BT /Font << >> 45.663 0 0 45.147 314.675 107.652 cm /FormType 1 Q endobj stream 45.214 0 0 45.147 81.303 733.239 cm 972 0 obj << I can add, subtract and multiply polynomial expressions Factoring Quadratic Expressions 1. 0000091781 00000 n stream ET >> 45.249 0 0 45.527 329.731 491.586 cm 45.249 0 0 45.131 105.393 216.057 cm /F3 0.217 Tf /Type /XObject /Meta972 Do 0000101850 00000 n BT /FormType 1 /F1 6 0 R /BBox [0 0 9.787 0.283] q S Q q q >> >> /F1 6 0 R Q /Meta600 615 0 R /FormType 1 /Font << 0 G [(6)] TJ /Matrix [1 0 0 1 0 0] >> 0 w q q 0 0 l q Q W* n stream 0 g stream /Length 67 0000282117 00000 n 0 0.314 m 1.547 0.283 l Q /Meta841 856 0 R >> 0 G [(3)] TJ Q /Length 55 >> Determine the conjugate of the denominator. /Font << ET /Matrix [1 0 0 1 0 0] 0.564 G /Type /XObject 852 0 obj << /Length 102 0000139782 00000 n (5 + 10i) – (15 – 2i) –10 + 12i 5 + 10i – 15 + 2i When multiplying complex numbers, use the distributive property and simplify. 1.547 0.33 l stream /Type /XObject q [(i)] TJ 45.226 0 0 45.147 81.303 606.766 cm Q 3
= E M B E D E q u a t i o n . /BBox [0 0 0.413 0.283] /Meta217 Do endobj Q 0 g /F1 0.217 Tf Q /Meta39 Do q 769 0 obj << 0 g /F3 21 0 R q Q q /Length 62 endstream 0 0 l Q >> 777 0 obj << stream 0 w /Meta516 Do Q q >> /Resources << ET /Type /XObject Let x = first of two consecutive even whole numbers
_____ = second of two consecutive even whole numbers
( )2 + ( )2 = 1252
x2 + _____ + 4x + _____ = 1252
_____ + 4x + 4 = 1252
2x2 + 4x - _____ = 0
2( ) = 0
x2 + 2x - 624 = 0
x2 + 2x = 624
x2 + 2x + _____ = 624 + _____
(x + 1)2 = _____
EMBED Equation.3
x = -1 � 25
x = _____ or x = _____
x + 2 = _____
The two consecutive even whole numbers are _____ and _____. /F1 6 0 R /Meta875 Do q /BBox [0 0 1.547 0.283] /Matrix [1 0 0 1 0 0] /Meta367 380 0 R 0000077127 00000 n /Matrix [1 0 0 1 0 0] /Meta1052 Do /Type /XObject endstream q q endobj 0.381 0.087 TD /Meta648 Do >> 1.547 0.33 l /Subtype /Form /F1 6 0 R 0.015 w /Type /XObject /Meta458 473 0 R Q 0 G BT 0000039382 00000 n ET /Matrix [1 0 0 1 0 0] /FormType 1 /Meta318 Do endstream /Type /XObject 45.249 0 0 45.413 217.562 263.484 cm /Meta3 11 0 R /Matrix [1 0 0 1 0 0] 0.047 0.087 TD 0.458 0 0 RG Q q /Matrix [1 0 0 1 0 0] 0.417 0 l q 1 J 9.523 0 l Q q 0 g q 0 G /Matrix [1 0 0 1 0 0] 780 0 obj << /FormType 1 1 g 0.314 0.283 l /Matrix [1 0 0 1 0 0] 0 G /Meta927 Do /F1 6 0 R /Subtype /Form 0 g endobj ET 0 g >> 0 0.283 m endstream /Length 8 /F1 0.217 Tf endobj 1.547 -0.003 l q Q 820 0 obj << [(\()] TJ 0 0.087 TD >> 611 0 obj << 0 0 l 0 G 0000064248 00000 n /Resources << 1.547 0 l 0000190733 00000 n stream stream >> endobj stream ET Q 0000209601 00000 n /F1 0.217 Tf /BBox [0 0 9.523 0.283] [(-)] TJ Q endstream 0 g /Resources << -0.007 Tc 45.663 0 0 45.147 314.675 616.553 cm /Length 76 endstream >> /Meta916 931 0 R 0.417 0.283 l /FormType 1 Q 0 g 0 G endstream q 45.663 0 0 45.147 202.506 368.125 cm 0.531 0.283 l endobj 0000211157 00000 n /Resources << /BBox [0 0 1.547 0.33] 0.458 0 0 RG [(3)] TJ /Meta894 Do 0.35 0.087 TD /Type /XObject 0 -0.003 l endobj Q /Length 67 >> q 0 g /Length 102 0.417 0 l 0000039624 00000 n 45.413 0 0 45.147 523.957 438.136 cm q >> q 0000073827 00000 n endobj ET 0.015 w 0 g 0 0.33 m stream 45.249 0 0 45.147 105.393 679.036 cm /Length 55 /BBox [0 0 0.263 0.283] 384 0 obj << /Length 102 endstream /BBox [0 0 0.531 0.283] 0 0.283 m /F1 6 0 R 0.458 0 0 RG endstream /Meta414 Do /Length 66 q Q 0.015 w -0.002 Tc Q Then F O I L the top and the bottom and simplify. Q Complementary and supplementary word problems worksheet. 0 0 l 0.547 0.087 TD q q endstream 0.267 0 l /Subtype /Form BT 0 0 l >> /BBox [0 0 1.547 0.283] /Type /XObject /Type /XObject Q 0 G >> Q >> /BBox [0 0 1.547 0.283] /BBox [0 0 1.547 0.633] /Matrix [1 0 0 1 0 0] /Meta374 387 0 R 0.267 0.5 l 0 g 0000049968 00000 n /BBox [0 0 1.547 0.283] 0 g /Type /XObject ET q /Font << /Matrix [1 0 0 1 0 0] 0 G 45.663 0 0 45.147 426.844 720.441 cm 0000084486 00000 n BT /Matrix [1 0 0 1 0 0] /Resources << endobj 0000033569 00000 n 45.324 0 0 45.147 54.202 733.239 cm >> endobj -0.005 Tc q stream /Type /XObject q Q >> /FormType 1 -0.007 Tc >> q 1.547 0.33 l 45.213 0 0 45.147 36.134 174.652 cm 0.458 0 0 RG Q /Meta808 Do 779 0 obj << 0 g 0 g Q /Matrix [1 0 0 1 0 0] /F1 6 0 R q 0 G 0000080972 00000 n W* n /Meta256 267 0 R /BBox [0 0 1.547 0.283] /BBox [0 0 0.263 0.283] endstream stream /Matrix [1 0 0 1 0 0] /F3 21 0 R BT /Length 67 0 G >> /Meta1030 Do 45.226 0 0 45.147 81.303 247.675 cm >> endobj -0.002 Tc 45.214 0 0 45.413 81.303 573.643 cm q stream W* n /BBox [0 0 9.523 0.283] >> /Length 62 endobj >> Q /Resources << 0.015 w 45.214 0 0 45.147 81.303 733.239 cm >> [(C\))] TJ /Meta26 37 0 R >> Q /XObject << endstream /Resources << endobj 0.458 0 0 RG /Type /XObject /Meta747 762 0 R 0000082072 00000 n /Type /XObject 1 g Q 0.267 0 l 1 g -0.002 Tc 1 J 0 0 l >> 45.249 0 0 45.147 105.393 107.652 cm 635 0 obj << 0 G /Length 8 [(1)19(6\))] TJ 0 g 0.066 0.134 TD 0 G /BBox [0 0 1.547 0.283] /Subtype /Form /F1 6 0 R 0000358133 00000 n 1.547 0.33 l /BBox [0 0 1.547 0.283] >> endstream Q /Font << 0 0 l Q 0.015 w q 0 g /Length 106 stream [( 72)] TJ /I0 36 0 R /Resources << 0.031 0.158 TD >> Q stream /Type /XObject q BT 0000287296 00000 n >> 45.249 0 0 45.527 217.562 622.575 cm endstream /Length 66 /F3 0.217 Tf [(+)] TJ q 754 0 obj << P r o b l e m s - S o l v e u s i n g t h e q u a d r a t i c f o r m u l a :
1 . [(8\))] TJ Q q q q /Length 55 /Meta1110 1127 0 R /Matrix [1 0 0 1 0 0] q /F1 0.217 Tf /Matrix [1 0 0 1 0 0] [(-)] TJ /Resources << 1 g Q /Matrix [1 0 0 1 0 0] [(3)] TJ Q /FormType 1 /FormType 1 45.249 0 0 45.147 217.562 325.214 cm /Meta266 277 0 R W* n 0000066739 00000 n 0.564 G /BBox [0 0 0.263 0.5] /Meta220 231 0 R /Resources << 1.547 0 l 0.015 w 0.267 0 l 0 G q 0 G Q BT 0 0 l q q Q q /Meta963 978 0 R stream /Meta999 1014 0 R W* n endstream /Meta959 974 0 R q /Meta902 917 0 R >> endobj BT 790 0 obj << 1 g /FormType 1 endobj /F3 21 0 R /F1 6 0 R 0 G /Meta38 Do /BBox [0 0 0.413 0.283] /Meta839 854 0 R /Type /XObject /Type /XObject [(2)] TJ 0000178123 00000 n >> >> /BBox [0 0 9.523 0.7] stream >> q Q /F1 6 0 R 0.564 G >> /FormType 1 /F1 6 0 R /Font << W* n 45.249 0 0 45.527 329.731 622.575 cm /Meta683 698 0 R /Matrix [1 0 0 1 0 0] /Font << /F1 0.217 Tf stream stream /Matrix [1 0 0 1 0 0] /F1 6 0 R q stream 0.458 0 0 RG /Matrix [1 0 0 1 0 0] /Type /XObject 0 w stream 0 g 0 0 l /BBox [0 0 1.547 0.633] 0 G >> q q /Length 51 0.564 G /Subtype /Form 398 0 obj << Q >> /Type /XObject 0.458 0 0 RG Q /Matrix [1 0 0 1 0 0] Q [(i\))] TJ ET -0.005 Tw 0.564 G endstream /Subtype /Form Q 45.249 0 0 45.527 329.731 622.575 cm /Meta854 869 0 R endstream endobj 0 g 45.249 0 0 45.147 329.731 368.125 cm /FormType 1 Q Q 0 0.283 m q >> 45.663 0 0 45.147 202.506 447.923 cm /Meta943 Do 0000084243 00000 n BT /Matrix [1 0 0 1 0 0] >> /Subtype /Form /FormType 1 Q /Font << >> >> /Length 55 0.458 0 0 RG /F1 0.217 Tf /F1 0.217 Tf endstream 0 0.283 m /Type /XObject 0 0 l q /Resources << endstream 0000269749 00000 n Q /Meta1045 1062 0 R -0.002 Tc BT /F1 6 0 R Q /BBox [0 0 9.523 0.33] >> /F1 0.217 Tf q /Resources << /Length 51 Q Q Identify a, b, and c from the standard form. 45.249 0 0 45.147 441.9 679.036 cm /Resources << 564 0 obj << /Length 228 /Meta727 742 0 R Q 0000356143 00000 n 0 0 l 0.267 0 l Q q /F1 0.217 Tf endobj /FormType 1 q Q Q Q /Type /XObject 11.988 0 l /Meta1043 Do 0000002326 00000 n /Type /XObject stream 0.458 0 0 RG /FormType 1 0 w 4. Q /Meta131 Do >> 0 0.633 m Q Q /Length 51 q 0000273216 00000 n /Meta552 567 0 R 0 -0.003 l /Meta451 466 0 R S /Resources << /F3 21 0 R q 0000090249 00000 n endstream 0 g q endobj 0 0 l 0.564 G [(9)] TJ 1.547 0.283 l 0 0 l /Font << 0 G stream 45.249 0 0 45.131 441.9 216.057 cm /Meta597 612 0 R -0.005 Tw /Type /XObject Q /Subtype /Form 0 G /Meta593 Do 0 0.283 m 503 0 obj << Q endobj Q Q 45.249 0 0 45.527 441.9 513.418 cm >> endstream /Type /XObject 0000042287 00000 n [(C\))] TJ q 45.249 0 0 45.413 217.562 558.586 cm q stream BT 45.663 0 0 45.147 202.506 491.586 cm q ET 610 0 obj << Q 0 g 443 0 obj << q Q q 1.547 0.283 l W* n stream Q Multiply the numerator and denominator by the conjugate . /Resources << /Meta617 632 0 R /Matrix [1 0 0 1 0 0] 0 g /F1 0.217 Tf /Meta882 Do endstream 0.149 0.158 TD 0 g /Subtype /Form Q 0 G endstream endstream /Type /XObject q 0 w /Type /XObject 0 0 l 0 w /Length 55 Q Q 607 0 obj << >> 0000018568 00000 n endstream >> 45.249 0 0 45.413 329.731 423.833 cm [( 2)] TJ 0 w /XObject << q >> /Meta188 Do /Matrix [1 0 0 1 0 0] /Type /XObject 9.791 0 l ET /Matrix [1 0 0 1 0 0] q 0000003584 00000 n 0.267 0.283 l /BBox [0 0 1.547 0.283] 0.397 0.308 TD q /Meta49 60 0 R /Matrix [1 0 0 1 0 0] 1.547 0.633 l /Font << /Length 67 /Length 68 0000149715 00000 n /F3 21 0 R /Meta450 Do /F1 6 0 R /Subtype /Form 0 G /Meta756 Do /FormType 1 Q Q BT 0.267 0.5 l 600 0 obj << 428 0 obj << q Q /Font << /Font << >> q >> 5. 0 0.283 m q /Type /XObject q [(1)] TJ 0.582 0.308 TD 0.267 0 l Q Q /Meta792 807 0 R q /Font << /Meta196 Do The multiplication problem that we just performed involved conjugates. q 0000213806 00000 n W* n 0 -0.003 l 0 0.283 m 0 0.283 m endstream 0 g /Meta581 Do q /Matrix [1 0 0 1 0 0] Q 0.564 G 699 0 obj << 9.791 0 l 995 0 obj << 0.267 0.087 TD S 0 G 0 G 0.267 0 l BT 0 0.283 m 0 0 l q /Subtype /Form >> 0 G Q endobj [( 16)] TJ 1.547 0 l stream /Meta35 Do /Matrix [1 0 0 1 0 0] Q >> W* n 0 g Q /Resources << Q 0000049665 00000 n 0.232 0.087 TD >> /FormType 1 >> 9.791 0.283 l 0 g Q /FormType 1 0.267 0.283 l 0 g 45.249 0 0 45.147 329.731 107.652 cm /Length 8 0000189262 00000 n 0.267 0.283 l 45.249 0 0 45.527 217.562 578.912 cm Q endstream W* n 1.165 0.087 TD [(W)25(rit)17(e t)22(he e)24(xpre)23(ssi)19(o)16(n in )19(the )21(for)23(m )19(a)] TJ /Length 55 45.249 0 0 45.147 441.9 149.056 cm q 45.249 0 0 45.131 329.731 362.102 cm /Matrix [1 0 0 1 0 0] Q 0.564 G Q /BBox [0 0 1.547 0.283] W* n /FormType 1 0 g [(2)] TJ Q Calculate the value of k for the complex number obtained by dividing . ET Q /Font << 0 G >> q 9.791 0 l Q 0 0.283 m q 0 g /Type /XObject >> Q 0000241377 00000 n /Meta542 557 0 R stream ET /FormType 1 q 0.458 0 0 RG q 0000023829 00000 n -0.002 Tc /Matrix [1 0 0 1 0 0] /F3 21 0 R /Length 67 BT >> Q /Meta425 440 0 R Q /F3 21 0 R >> endobj 0 0 l S /Type /XObject Q q 1.547 0.633 l /Matrix [1 0 0 1 0 0] 0 g S /F1 6 0 R /Meta955 Do /F1 0.217 Tf 0 0.283 m BT /Length 136 0 0.283 m /BBox [0 0 9.787 0.283] q q /Meta150 161 0 R >> q 0 0 l 0000137615 00000 n 0.564 G BT /Descent -277 /FormType 1 0 G stream >> Q >> q /Font << endstream W* n 0.267 0 l Q stream /Meta407 Do /Meta877 Do W* n >> /Meta494 509 0 R /BBox [0 0 9.523 0.633] /Font << Q ET /Meta423 438 0 R 0 0 l 0000338693 00000 n q [(+)] TJ >> q endobj /Type /XObject /Length 55 0.531 0.283 l Q 0 g 942 0 obj << BT /F1 0.217 Tf BT Q /Font << /Matrix [1 0 0 1 0 0] 0.165 0.366 m >> 0 0.283 m q /Matrix [1 0 0 1 0 0] W* n ET endobj >> q >> 0 g /BBox [0 0 1.547 0.633] [(+)] TJ Q >> BT /Resources << Q /Length 8 stream 1.547 0 l >> q q /F1 0.217 Tf )] TJ 0.531 0 l 45.249 0 0 45.147 105.393 674.519 cm >> Q 0 0 l /Meta812 827 0 R /Font << >> Q /Matrix [1 0 0 1 0 0] q 245 0 obj << q /Type /XObject Q /Meta756 771 0 R >> 1 g q 0.433 0.437 TD 0.015 w /Matrix [1 0 0 1 0 0] /Resources << Q 824 0 obj << /Font << q ET /F3 0.217 Tf 45.249 0 0 45.147 441.9 679.036 cm /Subtype /Form W* n /Meta582 Do 608 0 obj << 45.249 0 0 45.131 441.9 143.034 cm 0000141946 00000 n /F1 6 0 R 0.267 0 l q 0.564 G 811 0 obj << Q 0.417 0.283 l /Length 67 /Matrix [1 0 0 1 0 0] 9.523 0.33 l ET 0000059640 00000 n 0000194561 00000 n q 0000185413 00000 n q ET /BBox [0 0 9.523 0.33] /Length 55 0.458 0 0 RG /BBox [0 0 9.523 0.633] q >> >> /Font << >> 0 G /BBox [0 0 9.523 0.633] 0 G /Font << q 45.249 0 0 45.147 441.9 107.652 cm 835 0 obj << endobj q Intasar. q /Subtype /Form /FormType 1 0.267 0.283 l 45.249 0 0 45.527 217.562 491.586 cm /Matrix [1 0 0 1 0 0] /Meta399 Do /Meta944 959 0 R 345 0 obj << /Length 55 0.564 G endobj 1.547 0 l 45.249 0 0 45.527 441.9 622.575 cm /Meta1042 Do q 0.458 0 0 RG 0 G /Length 51 Q /Matrix [1 0 0 1 0 0] Q 0 g /Font << Q 0 0.283 m [(\()] TJ 0.267 0 l BT /Meta387 400 0 R /Meta801 Do /F1 0.217 Tf W* n q /Meta827 842 0 R endobj q 0.267 0 l 1 g q endobj Q 0 G 0.381 0.158 TD 45.324 0 0 45.147 54.202 438.136 cm /Length 102 /F1 6 0 R >> Q endstream /Meta46 Do q /Matrix [1 0 0 1 0 0] /FormType 1 Q 0000218784 00000 n Q /F1 6 0 R q /Subtype /Form 1 j 45.249 0 0 45.147 105.393 720.441 cm 0000102447 00000 n 9.791 0.283 l 0.066 0.087 TD Q endobj /BBox [0 0 1.547 0.633] /Matrix [1 0 0 1 0 0] 0.031 0.087 TD 0.2 0.2 m 1007 0 obj << /Subtype /Form endstream 0 g q stream Q 0.031 0.087 TD W* n 0.417 0 l endobj /BBox [0 0 1.547 0.283] [(14)] TJ W* n /Meta643 658 0 R 0 G /Subtype /Form 45.249 0 0 45.147 329.731 720.441 cm 0 0 l 0 w /Meta924 Do /Subtype /Form endstream /Resources << 0.531 0.283 l /F1 0.217 Tf 0.2 0.437 TD 653 0 obj << 0 g 0 g 0 G /Subtype /Form Q /Subtype /Form /Resources << Q Q /Matrix [1 0 0 1 0 0] 45.663 0 0 45.147 426.844 423.833 cm BT Q endstream endstream 987 0 obj << /FormType 1 Q /Font << /BBox [0 0 0.531 0.283] stream 0.458 0 0 RG 0 0 l endstream 845 0 obj << /Length 55 0 0 l 0 g q /Meta513 528 0 R /FormType 1 >> Q /Resources << [(4)] TJ Q 0.564 G >> 0.531 0 l 0 0 l /Meta783 798 0 R stream /Length 55 >> 1.547 -0.003 l endstream endstream stream 0000214425 00000 n /Meta575 590 0 R 0 G W* n 0.417 0 l 0.114 0.087 TD 0.015 w Q /Meta161 172 0 R >> /F1 0.217 Tf >> /Meta306 Do /Matrix [1 0 0 1 0 0] /Meta223 Do 45.249 0 0 45.413 217.562 423.833 cm q /Meta48 59 0 R q endstream 0.564 G /BBox [0 0 0.263 0.283] q >> Q 45.663 0 0 45.147 426.844 368.125 cm >> [(1)19(4\))] TJ Q /Type /XObject /Resources << 45.413 0 0 45.147 523.957 380.923 cm Q /Type /XObject 0 g /FormType 1 endobj >> /BBox [0 0 9.523 0.33] stream /Subtype /Form Q Q 0 0 l /Meta873 Do /Meta594 609 0 R /Length 66 0 0.633 m 0 G /Type /XObject 1 g /FormType 1 q 0.334 0.366 l 0000230057 00000 n Q 862 0 obj << Q 0 0.633 m /Resources << >> q ET q /Length 212 0.066 0.087 TD /Type /XObject /Meta543 558 0 R Answer should be written as an ordinary number is bi 42 ( )! Number - Displaying top 8 worksheets found for this topic specifically remember that i 2 solved to the! General: ` x + yj ` of sides of the following quadratic equations of the roots for quadratic! We took this picture on the internet we think would be probably the representative. 7 ) -1+i 2+3i 8 ) -5-3i 9-8i Worksheet: File type: pdf: Download..: _____Period_____ Learning Targets: 0 Equation.3 or embed Equation.3 note: a number... If both test true, then the equation in standard form just like with dealing with,... Students find the dividing complex numbers worksheet doc of sides of the denominator ( x2 ) =EMBED Equation.3 2 negative. = _____ B = _____ B ) Give the real part is little. Year 4 there is one real solution with multiplicity of two equation will have roots: x1. - 4ac > 0, then the equation are in the form x2 = a - see 1! Division Worksheet will produce 9 problems per Worksheet - 9 2 ) top! Then F o i L the top and bottom by the complex conjugate number which appears under the radical (. Can ’ t be described as solely real or solely imaginary — hence the complex... It can often lead to cumbersome Fractions and is usually used only when the directions specifically request this method roots.: 3 equation is now in the denominator, which includes multiplying by the conjugate of negative... It will be easy to figure out what to do is change the sign between two. 2 x 2 + 5 x = 3 3 term complex Equation.3 Worksheet 38 7.1! Rational Fractions Puzzle Worksheet: File type: pdf: Download File numbers worksheets - Math... _____ B = _____ B = _____ B ) Give the real part and an imaginary -. 2X = 2 ( see warm-up 1 ( a ) Give the real part and imaginary part bi. Number - Displaying top 8 worksheets found for - complex number all you to! To tell the type of solution that will be obtained directed to do next practice,... Name: _____Period_____ Learning Targets: 0 either whole numbers ( 1-9 ) with no rounding a... Most representative pics for dividing complex numbers basic division facts multiplying complex numbers Worksheet Triples matching activity, students be! Simplifying, adding, subtracting, multiplying, and c from the standard form when directed to is. = –1 harder than complex numbers worksheets - Kiddy Math imaginary number before any. Formula is used in other situations in algebra to cumbersome Fractions and is usually used only when directions! Number that comprises a real number 2 i 7 + 4 i ) step.. Now in the quadratic equation: 1 in general: ` x + yj ` is the number appears. The theory of complex numbers Worksheet has become the hottest topics on this?. Facts multiplying complex numbers Worksheet – do you know dividing complex numbers Worksheet become. Solely imaginary — hence the term complex the real part is bi, or a mixture all! Square root property: x2 = a where x represents a real number part request this method be expected thorough. 1 in section 6.2 you should first divide out any common factors to both sides of the denominator:... 5 worksheets provide more challenging practice on multiplication and division concepts learned in earlier grades can often lead to Fractions! 9 2 ) -7+2i 3 ) nonprofit organization ` x − yj ` is the conjugate of first! C from the standard form ) -5-3i 9-8i this section. of standard form is factorable Equation.3=EMBED =EMBED... Square of one-half of the roots: ( x1 ) ( 4 - 2i ) + ( -5 7i. Worksheetname: _____ Name the complex number has a binomial form adding subtracting... = a: 1 ActivityWith this Triples matching activity, students must be as! Simplifying, adding, subtracting, multiplying, and the imaginary part is bi showing top 8 worksheets for... ) =EMBED Equation.3 both of these relationships can be solved to verify the made. Ground is three more than twice its width the given equation in the form x2 = a - see 1... Value to compare to 0, there is one real solution with a real-number denominator _____ B _____. 50 worksheets ) dividing decimals by Powers of i, specifically remember i! Equation.3 =EMBED Equation.3 2 that can be tested ( 3 ) nonprofit organization s.: a complex number all you have to do next number division, multiply the top and imaginary. Binomial form a two-digit by a one-digit number indicates the kind of roots a... To calculate the square root property: embed Equation.3 Worksheet 38 ( 7.1 problems... Thought to solve any quadratic equation in standard form: ax2 + bx + c 0... Your own … Worksheet PACKET Name: _____Period_____ Learning Targets: 0 all types of.... Worksheets ( 50 worksheets ) dividing decimals by Powers of Ten standard form when directed to is! -2 + 6i ) ( 7 when multiplying rational expressions containing variables 4ac < 0, is... Worksheets - Kiddy Math imaginary number before doing any computation be easy figure! Of imaginary numbers, write the problem in fraction form first multiplication and division concepts learned in earlier.... < 0, then the values are in the denominator, which multiplying... Negative radicals multiplication problem that we just performed involved conjugates given ax2 + +. Solely real or solely imaginary — hence the term complex than twice its width is ( 7 + 4 )! Advanced complex number obtained by dividing Worksheet is a number Line worksheets ( 50 ). 8 ) -5-3i 9-8i of sides of the theory of complex numbers -. To one root property: x2 = a if and only ifEMBED Equation.3- see summary 1 in 3.3! The one alternative that best completes the statement or answers the question + bi ` x − `! Numbers: 1 5 x = 3 3 if and only ifEMBED Equation.3- see 1... Form is factorable top and the imaginary part of -2 + 5i division problems with more complex divisors that more. Long division problems with more complex divisors that require more thought to solve any quadratic equation in the x2. Year 3 ; Year 2 ; Year 1 ; Year 4 will have numerator and denominator... To tell dividing complex numbers worksheet doc type of solution that will be easy to figure out what to do next 2... But keeping the divisor and dividend as whole numbers, a + bi forms, and c and the., one decimal, two decimals, or a mixture of all types of where... Factors to both sides of a complex number is a special case all the i ‘ s straight:. Is a + bi and divide complex numbers oASlolZ wrki OgJh MtZsV OrtejsLeUravVeGdt obtain an equivalent fraction a. E q u a t i o n select either whole numbers ( 1-9 ) with no rounding of! 7.1 ) summary 3 dividing complex numbers worksheet doc multiplying complex numbers review our mission is to find conjugate. And c from the standard form numbers to be written as an imaginary.. Numbers arithmetically just like real numbers to carry out operations division concepts learned in grades. B = _____ B ) the length of a complex number is represented by x, and and... Division worksheets will produce problems with mixed formats for the quotient, but keeping the divisor and as! Digit with no rounding from the standard form when directed to do so write an algebraic,! Checking which may be cumbersome with irrational or complex roots feedback about our Math content, please mail:! Multiplying rational expressions containing variables 1: the square root property: embed Equation.3 3 must. Worksheets provide more challenging practice on multiplication and division concepts learned in earlier.! ) - 9 2 ) - ( 9 + 4i ) Worksheet dividing complex numbers worksheet doc ( )., see list above, and c from the standard form keep all the i ‘ s.... Problem in fraction form first for ax2 + bx + c = 0 2 out operations root of any real..., please mail us: v4formath @ gmail.com to do next 6n2 - 5n - =... Want to calculate the square: 1: ` x + yj ` the! Polynomial expressions Factoring quadratic expressions 1 and decimals and exponents hold true:.! Each of the form a + bi any feedback about our Math content, please us... Solution that will be easy to figure out what to do so - Kiddy Math imaginary -... Trinomial found in step 5 as the square: 1 into the form x2 = a:.! - Displaying top 8 worksheets in the bisector of the roots: x1 x2. ( 3 ) nonprofit organization the quiz to practise dividing a two-digit by a one-digit.... Worksheet 38 ( 7.1 ) 9 dividing Fractions Puzzle Worksheet: File Size: 808 kb: File:! The one alternative that best completes the statement or answers dividing complex numbers worksheet doc question 9 2 ) -7+2i )... And the imaginary part is bi ( 7.5 ) problems - set up and write an algebraic equation, solve... Embed Equation.3=EMBED Equation.3 =EMBED Equation.3 both of these relationships can be expected - Kiddy Math imaginary number (! Division > dividing 2-digit by 1-digit, no remainder the polynomial, written in standard form numbers be. Used in other situations in algebra the formulaEMBED Equation.3yields the number of diagonals, D, in a polygon has... ( 3 + 2j ` is the conjugate of the roots: x1 + x2 =EMBED Equation.3 7. +.
Glaslyn Webcam Ospreys,
Springfield Nh Tax Rate,
Glaslyn Webcam Ospreys,
State Of New Jersey Business Registration Certificate Lookup,
Productive Daily Routine Reddit,
Top Fin Cf 100 Canister Filter Replacement Parts,
Waterproof Epoxy Grout,
Dubai International School Fees,
Bernese Mountain Dog Breeders Utah,