stream 45.249 0 0 45.147 441.9 325.214 cm 0.015 w q W* n /Meta569 Do q /Meta369 Do 0000003584 00000 n Q 45.214 0 0 45.131 81.303 244.664 cm Q 45.249 0 0 45.147 105.393 720.441 cm Q 0000206328 00000 n >> 0 0.283 m 0000044730 00000 n /Type /XObject q 0.531 0.283 l q 0000044964 00000 n 0.458 0 0 RG /Meta474 Do 0 0.283 m 0 0 l ET /F1 0.217 Tf 0 g /Font << /BBox [0 0 1.547 0.283] 0.002 Tc q endobj 0 g /Meta1063 Do 0 g 0 g /Font << /Meta299 312 0 R /Length 55 0.2 0.165 l 296 0 obj << q BT Q /BBox [0 0 1.547 0.33] 0.031 0.437 TD /F1 6 0 R 0.334 0.299 l 0000016059 00000 n q 0.031 0.087 TD q 45.249 0 0 45.147 329.731 447.923 cm >> /FormType 1 /Resources << 0 G These section presents rounding worksheets for rounding whole numbers and rounding decimal numbers starting with relatively simple problems that introduce the rounding algorithm and then advance to more complex problems where students must determine the correct place value digit to check as well as the correct digit to round up or round down.. >> 45.663 0 0 45.147 426.844 679.036 cm BT Q 0 G Q 0000170054 00000 n 0 G It allows for looking ahead to tell the type of solution that can be expected. >> 0 0.283 m q /Resources << q q /Meta813 Do /Matrix [1 0 0 1 0 0] /FormType 1 /BBox [0 0 0.314 0.283] Q q >> W* n /Matrix [1 0 0 1 0 0] /F3 0.217 Tf /Subtype /Form /Subtype /Form /Matrix [1 0 0 1 0 0] /Meta371 384 0 R Q 0 G q 1 J /Matrix [1 0 0 1 0 0] /Matrix [1 0 0 1 0 0] 832 0 obj << 45.663 0 0 45.168 426.844 216.057 cm Q /F1 6 0 R 969 0 obj << 446 0 obj << 45.249 0 0 45.527 329.731 578.912 cm 0 w endstream /Length 55 /F1 0.217 Tf 0 g 603 0 obj << Q BT q Q >> BT /XObject << 0.417 0.283 l q >> endobj BT 0.267 0 l /Length 55 /Meta169 180 0 R q >> endobj q /Matrix [1 0 0 1 0 0] /Matrix [1 0 0 1 0 0] >> -0.005 Tw >> >> /FormType 1 /Meta319 Do /BBox [0 0 1.547 0.633] /F1 0.217 Tf 0 g endstream Q /Subtype /Form /Meta423 438 0 R >> /Matrix [1 0 0 1 0 0] Q q Q 0 0.633 m Q Q 0 g /Length 51 0 0 l /FormType 1 0000174740 00000 n /Length 55 Find, to the nearest tenth, the length of one side of a square with a diagonal that i s 2 0 m e t e r s l o n g . q 0 0.283 m >> Q /BBox [0 0 9.523 0.633] /Meta505 Do /BBox [0 0 1.547 0.33] Q W* n >> 0 G 0.417 0 l 0.564 G endobj /F1 6 0 R /Type /XObject >> 1 J /Meta782 Do /F1 0.217 Tf /Length 303 stream /Type /XObject q 0 0.087 TD >> /F3 0.217 Tf >> Q ET 0.334 0.087 TD q /Meta379 392 0 R /Type /XObject /Type /XObject 0 G 0.015 w >> endobj Q q Q Q 45.249 0 0 45.131 105.393 289.079 cm /Subtype /Form 362 0 obj << Q q Q q q Q Q BT /Meta618 Do /Resources << S /F1 0.217 Tf /FormType 1 Q Q /BBox [0 0 1.547 0.633] endstream >> /F1 0.217 Tf /Length 55 /Meta1005 Do Q /Subtype /Form Q /Meta961 976 0 R 45.249 0 0 45.527 217.562 535.249 cm 0000097505 00000 n Q 0.114 0.087 TD ET 45.324 0 0 45.147 54.202 687.317 cm 0 0.283 m 0000242937 00000 n Q 0.458 0 0 RG endstream q q Q q /FormType 1 >> /Meta267 278 0 R 0.531 0 l ET /Type /XObject 0 g >> endstream 0 g 528 0 obj << W* n /Length 65 /Type /XObject Q 0 g 0.458 0 0 RG BT 0 0 l /Resources << 0000210317 00000 n q endstream 1028 0 obj << /Meta619 634 0 R /F1 6 0 R q Learn. /Meta834 849 0 R /Subtype /Form Q stream q 0000082307 00000 n 729 0 obj << 0 0.283 m 757 0 obj << W* n 0.267 0.283 l BT q /Resources << 0 w 447 0 obj << 0000234812 00000 n 0.564 G /Font << stream Q 0.433 0.437 TD /BBox [0 0 9.523 0.33] /Matrix [1 0 0 1 0 0] q /Font << q endobj q 0 G >> /Length 102 /F1 6 0 R Q q 45.324 0 0 45.147 54.202 730.98 cm Q 0000189262 00000 n /FormType 1 0000276103 00000 n >> 0 G q Q /Meta549 Do /Resources << 0.267 0 l 1091 0 obj << [(97)] TJ 675 0 obj << endstream >> 0000224790 00000 n /Matrix [1 0 0 1 0 0] 0 G /Subtype /Form Q >> 45.249 0 0 45.147 441.9 447.923 cm /FormType 1 0.267 0.5 l /FormType 1 q 0 G 0 G /F3 21 0 R >> q BT Q /Subtype /Form /Matrix [1 0 0 1 0 0] >> Q /BBox [0 0 1.547 0.633] 0.681 0.401 l Q 0.696 0.087 TD q 0000218404 00000 n endstream 0.2 0.158 TD 0000210075 00000 n Q /Type /XObject -0.002 Tc Q Q q >> q /F1 6 0 R W* n /Matrix [1 0 0 1 0 0] 0.031 0.154 TD q /Meta27 38 0 R BT 45.663 0 0 45.147 90.337 718.183 cm >> /FormType 1 >> Q 0.564 G >> W* n Q 0000289237 00000 n endstream -0.002 Tc endstream /F1 6 0 R ET /Matrix [1 0 0 1 0 0] 1.547 0.633 l stream /Matrix [1 0 0 1 0 0] 1 g ET 0.216 0.087 TD ET q W* n /Resources << 0000036989 00000 n Q /Length 66 /FormType 1 /Meta227 Do q q /Length 67 /F1 6 0 R Q >> 0 G 5. q 0 g /Meta206 217 0 R /Subtype /Form /F1 6 0 R q /Length 75 0 G endobj 0.267 0.087 TD q Q endstream /Type /XObject 0000252136 00000 n Q 0.015 w /Resources << 1.547 0 l q 0000194319 00000 n Q /FormType 1 /Meta325 Do /Subtype /Form 0.267 0.5 l Q 45.249 0 0 45.147 217.562 630.856 cm /Meta929 944 0 R >> ET Q ET >> /BBox [0 0 0.263 0.283] endobj /FormType 1 >> q Q 0 0.283 m Q >> >> q >> 0.564 G q 0 g >> 9.791 0 l stream /Meta961 Do /Subtype /Form /Resources << /F1 0.217 Tf 721 0 obj << /F1 6 0 R /Type /XObject /Meta20 30 0 R 0.35 0.299 l endobj /FormType 1 stream endobj S /Resources << endstream Q endobj 0.001 Tc q q /F1 6 0 R stream 0 g 0000200184 00000 n >> 9.791 0.283 l /Resources << 0.531 0 l /Type /XObject Note: When b2 - 4ac = 0, there is one real solution with a multiplicity of two. ET Q 317 0 obj << 0.015 w >> /BBox [0 0 1.547 0.283] W* n /F1 0.217 Tf /Type /XObject Q stream /Meta817 832 0 R 0 G Q /FormType 1 q 1 g Q /Resources << 0000040834 00000 n 0 G >> /Font << q /Matrix [1 0 0 1 0 0] stream stream stream >> 0.216 0.473 TD 1 g /Length 94 W* n /Font << q 1 g 0 0 l q 0.531 0.283 l /Matrix [1 0 0 1 0 0] Q T r y i t . /BBox [0 0 1.547 0.283] >> BT 0 0 l BT /Flags 32 endobj stream stream 0 g /Subtype /Form 45.663 0 0 45.147 314.675 107.652 cm 0.458 0 0 RG 464 0 obj << q q /FormType 1 0 g 0000340862 00000 n Q endobj /F1 0.217 Tf 262 0 obj << >> 0 g 0.458 0 0 RG 45.249 0 0 45.527 441.9 513.418 cm Q stream 0 g /Subtype /Form 0 G 532 0 obj << 0 G /Subtype /Form 45.249 0 0 45.527 329.731 578.912 cm >> /BBox [0 0 1.547 0.283] stream endstream q /Length 106 >> ET 1.547 0.283 l >> Q endobj Q /Meta144 155 0 R ET Q /Type /XObject … 983 0 obj << >> 45.249 0 0 45.527 441.9 622.575 cm stream 0.314 0.158 TD 0 g stream /BBox [0 0 1.547 0.283] 0 G 0.015 w /Meta205 216 0 R 0.458 0 0 RG 0 g /Subtype /Form /Meta112 Do BT stream >> /Meta181 192 0 R q /Meta179 190 0 R Q 0 0.283 m 0 0.283 m Complex Fractions Worksheet Hard. [(D\))] TJ /Subtype /Form Q q /Subtype /Form BT 0 g 0.381 0.087 TD /Type /XObject 0 0 l /Length 102 q /Subtype /Form 45.663 0 0 45.147 90.337 674.519 cm ET 279 0 obj << endobj q /Font << /Font << q >> endobj q Q /Matrix [1 0 0 1 0 0] q endstream stream BT /Type /XObject stream /Type /XObject >> /Resources << 0.015 w /Matrix [1 0 0 1 0 0] 0 g /Type /XObject >> q endobj 0 0 l 0.267 0 l If b2 - 4ac < 0, then the equation has two nonreal complex solutions. 0000019250 00000 n /Font << Q /Subtype /Form /Subtype /Form /Type /XObject /Matrix [1 0 0 1 0 0] q 0.015 w 0.5 0.366 m 0 g 0.417 0.283 l /Font << /FormType 1 ET /BBox [0 0 9.523 0.283] Q 0 G stream endstream >> q 1 g ET Q Q 0 g q >> 0000007940 00000 n >> 772 0 obj << /Meta596 Do 0.267 0.283 l /Type /XObject q /BBox [0 0 1.547 0.633] -0.002 Tc /Length 67 /Length 55 q W* n /Subtype /Form q q >> 847 0 obj << /FormType 1 Q 0000156244 00000 n >> /Length 94 >> 0 g /Matrix [1 0 0 1 0 0] 0000074298 00000 n /Subtype /Form /Meta44 Do /Subtype /Form [(i)] TJ Q Q 0 g Q Q 0000054333 00000 n stream 0.564 G q 0 G >> >> Q /Resources << endstream Q 0.598 0.158 TD /Font << /Subtype /Form 0 G /F1 0.217 Tf /Meta608 623 0 R q 45.249 0 0 45.527 217.562 578.912 cm stream [(2)19(1\))] TJ W* n q endstream 0 0 l /FormType 1 9.791 0 l 1 J Q 0 w [(i)] TJ Q 542.777 593.969 m /Subtype /Form /Matrix [1 0 0 1 0 0] endobj q stream /Meta715 Do endobj >> endstream BT stream /Meta659 Do 0.458 0 0 RG W* n /Subtype /Form /Meta899 914 0 R >> Q 1.547 0 l /Type /XObject 45.663 0 0 45.147 202.506 578.912 cm stream q 0000033802 00000 n endstream q /FormType 1 stream endstream 1 g 0 w 45.249 0 0 45.147 329.731 203.259 cm 0 g >> /F3 0.217 Tf /Meta829 Do endstream /FormType 1 endobj /F1 6 0 R /Length 8 >> Q 0.564 G 0000186585 00000 n /Length 55 Q /FormType 1 /F1 0.217 Tf q /F1 0.217 Tf /Meta1011 Do /Resources << /Subtype /Form >> endobj >> /Matrix [1 0 0 1 0 0] Q 0.267 0.283 l q 0.531 0.283 l /Matrix [1 0 0 1 0 0] endobj stream q /Meta460 475 0 R >> /F1 0.217 Tf /Type /XObject 1000 0 obj << 0.015 w /FormType 1 /Length 8 0 G 0 w 962 0 obj << 45.249 0 0 45.147 105.393 107.652 cm /Meta337 350 0 R stream /Font << >> Q >> /FormType 1 endobj /Length 55 >> /Meta764 Do /BBox [0 0 9.523 0.283] 0000034363 00000 n /Matrix [1 0 0 1 0 0] /FormType 1 0.564 G 0 0.283 m Q q /Meta471 Do 0000133389 00000 n 0.015 w >> W* n 1 g /Matrix [1 0 0 1 0 0] Worksheets: Dividing 2-digit numbers by one digit with no remainders. /Length 102 >> [(i)] TJ /Font << q 45.324 0 0 45.147 54.202 528.474 cm /Meta1082 1099 0 R q /Meta960 Do 45.663 0 0 45.147 202.506 368.125 cm 45.214 0 0 45.131 81.303 171.641 cm Q 0000212020 00000 n 0 g /FormType 1 /Meta1021 1036 0 R >> q q >> 753 0 obj << Q /Subtype /Form 0 0 l 0 g Two complex numbers . /Meta280 291 0 R 45.214 0 0 45.147 81.303 691.834 cm 1 g /Meta460 Do q /F1 0.217 Tf stream /Length 66 endstream endstream >> /Meta556 Do /BBox [0 0 0.263 0.283] ET W* n /F1 6 0 R /Matrix [1 0 0 1 0 0] q 1 j 554 0 obj << /BBox [0 0 1.547 0.283] /Matrix [1 0 0 1 0 0] [(-)] TJ 45.249 0 0 45.413 441.9 263.484 cm endobj Q 0 G 540 0 obj << >> BT q /F1 6 0 R 45.324 0 0 45.147 54.202 227.349 cm q 0 0.283 m 45.249 0 0 45.147 105.393 674.519 cm stream 0000253150 00000 n Q [(-)] TJ q q /Font << q /I0 Do 0 0 l Recall the quadratic formula:EMBED Equation.3 4. q >> [(12)] TJ /F1 6 0 R 0.564 G 45.249 0 0 45.131 329.731 143.034 cm Q 564 0 obj << 0.564 G 45.249 0 0 45.527 217.562 513.418 cm /Meta82 93 0 R /FormType 1 endobj /Length 51 /BBox [0 0 11.988 0.283] 0.165 0.087 TD Q 0000149483 00000 n q 45.249 0 0 45.131 105.393 216.057 cm /Length 67 0.015 w 45.249 0 0 45.131 105.393 216.057 cm >> endstream ET /F1 0.217 Tf 0 g 1.547 0.633 l Note: The product of a complex number and its conjugate is always a real number - see warm-up 3c and problem 9 in above summary. /Type /XObject /Subtype /Form 0 0 l 0 w >> 1.547 0.283 l Q [(-)] TJ /FormType 1 q /Type /XObject 424 0 obj << >> >> /Font << 45.249 0 0 45.147 441.9 447.923 cm >> >> 0 g q /F1 6 0 R 895 0 obj << 0 0.283 m Q /Meta186 197 0 R 0 0 l /BBox [0 0 0.263 0.283] q 0 G Q /BBox [0 0 9.523 0.283] /Font << /I0 36 0 R /Type /XObject Q 0 0 l 0.417 0 l 0.417 0.283 l Q 45.249 0 0 45.131 105.393 289.079 cm /Length 102 endstream ET /Subtype /Form 816 0 obj << >> 45.214 0 0 45.413 81.303 573.643 cm /F3 0.217 Tf 1 g stream >> /Matrix [1 0 0 1 0 0] /F1 0.217 Tf q /Matrix [1 0 0 1 0 0] 0.267 0 l /Subtype /Form /F1 6 0 R endobj /FormType 1 [(i)] TJ ET endstream Q 0.564 G /XObject << endobj 1 g /Subtype /Form [(2)19(0\))] TJ Q /Meta751 Do >> q /Meta47 Do 0 g Q stream endstream /Length 102 /Meta1098 Do 0.165 0.129 m /Subtype /Form Q 0.267 0.283 l /Meta23 33 0 R /Subtype /Form 223 0 obj << /Matrix [1 0 0 1 0 0] 0 G stream /F3 21 0 R /FormType 1 0 0 l Q 0000342302 00000 n 0.114 0.087 TD 920 0 obj << endstream ET q 45.214 0 0 45.413 81.303 380.923 cm /BBox [0 0 9.523 0.633] >> -0.001 Tc 1067 0 obj << W* n W* n Q 3 = E M B E D E q u a t i o n . /FormType 1 0.564 G /Meta590 605 0 R 2. /Matrix [1 0 0 1 0 0] /Subtype /Form >> BT 0.566 0.401 l 0.015 w endstream /Resources << 9.791 0 l 0 0.283 m 839 0 obj << endobj >> S -0.004 Tc 0 g /Meta15 25 0 R endobj >> 45.249 0 0 45.527 329.731 535.249 cm BT 0000351018 00000 n q /Meta735 Do /Type /XObject W* n q 538 0 obj << 0000257521 00000 n Practice this collection of printable worksheets and make headway dividing decimal numbers involving digits in the tenths’, hundredths’, and thousandths’ place by 10, 100, 1000, and so on! /Subtype /Form q 0 G 0.015 w ET stream 0000258933 00000 n /F3 0.217 Tf 1 j ET q /F1 6 0 R >> Q Q /Matrix [1 0 0 1 0 0] /Font << /F1 0.217 Tf endobj 0.881 0.087 TD >> [( 2)] TJ 45.663 0 0 45.147 90.337 447.923 cm >> q /F1 0.217 Tf 0 G >> Q /Type /XObject endobj 45.324 0 0 45.147 54.202 99.371 cm [(i)] TJ /F1 6 0 R Problems - Solve by completing the square: 3. /F1 0.217 Tf /Matrix [1 0 0 1 0 0] 747 0 obj << /BBox [0 0 0.263 0.283] >> Q /Type /XObject /BBox [0 0 0.263 0.283] 0000195173 00000 n /BBox [0 0 1.547 0.283] /F1 0.217 Tf 0 G 0 0 l [(-)] TJ q Q endobj /Matrix [1 0 0 1 0 0] 0 w /FormType 1 9.791 0.283 l /Type /XObject 0.814 0.087 TD Q >> /Resources << /Font << q 0000243083 00000 n Q /Resources << 0.267 0.087 TD 45.249 0 0 45.147 329.731 674.519 cm 0 g 0.031 0.087 TD /Resources << ET /Meta995 Do >> Examine the remaining terms on the left side to determine what value must be added to obtain a perfect square trinomial on the left side of the equation. endstream 0 w /Resources << >> W* n /Subtype /Form 0 G Q 0 G /FormType 1 /F1 0.217 Tf endstream endobj /Meta543 558 0 R 0 0.283 m 0 0.283 m endstream q q /Meta1085 1102 0 R >> /FormType 1 /Resources << /BBox [0 0 0.413 0.283] /F1 6 0 R 0 g endobj /Resources << /Resources << >> Q q 0.2 0.437 TD Q q Q 0 0 l /Meta275 286 0 R /Subtype /Form stream 337 0 obj << endobj (1 - 3i)(1 + 3i) Summary 4: Two complex numbers a + bi and a - bi are called conjugates of each other. 0.564 G /Font << 0000280304 00000 n /Matrix [1 0 0 1 0 0] Worksheet 41 (7.4) c) EMBED Equation.3=EMBED Equation.3 ; a = ____, b = ____, c = ____ EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 or EMBED Equation.3 EMBED Equation.3 o r E M B E D E q u a t i o n . /Length 102 /Meta249 260 0 R Q /F3 21 0 R /Meta22 32 0 R /BBox [0 0 0.263 0.283] 0 g q /FormType 1 45.663 0 0 45.147 202.506 622.575 cm /BBox [0 0 9.523 0.633] q endobj /Font << /Length 62 /Matrix [1 0 0 1 0 0] endobj /Resources << /Meta472 487 0 R 0 g Q /Subtype /Form /Meta998 Do /F1 0.217 Tf /Subtype /Form BT BT Q q Q 0 0 l 0000242099 00000 n endstream /Meta23 Do Q endobj 1.547 0.283 l 0.531 0.283 l Q /Matrix [1 0 0 1 0 0] /Meta747 Do 0 g 0 g 0 g 0 0 l 0.015 w /Resources << endstream Q 1.547 0.283 l /Length 55 >> /Length 55 q 0000161433 00000 n /FormType 1 0000196993 00000 n -0.007 Tc endobj /Meta730 Do 0.564 G 0 0.087 TD q 0000028236 00000 n /Matrix [1 0 0 1 0 0] q q /BBox [0 0 9.523 0.33] W* n endobj /Resources << endstream 45.214 0 0 45.131 81.303 390.709 cm 45.249 0 0 45.147 441.9 368.125 cm /Meta564 Do BT ET 0 G q /Font << Q 45.413 0 0 45.147 523.957 483.305 cm /Resources << 45.663 0 0 45.147 426.844 107.652 cm endobj 0.267 0 l Q 538.26 483.305 m Q 0 g 45.249 0 0 45.147 105.393 720.441 cm /Length 212 >> 0000231701 00000 n /BBox [0 0 1.547 0.283] BT 0 G BT S S q Q /FormType 1 BT Q 0 0 l q q BT Q >> /Meta274 Do /Matrix [1 0 0 1 0 0] /Meta555 Do /Meta766 Do /Resources << Note: A complex number has a binomial form. /Subtype /Form /Type /XObject 0.458 0 0 RG /Font << >> Q /F1 0.217 Tf q /Matrix [1 0 0 1 0 0] 9.523 0 l 45.249 0 0 45.147 105.393 447.923 cm 0 w 0.564 G 0.066 0.551 l 1108 0 obj << /Subtype /Form BT endstream 0 G 0000270068 00000 n 0 w /Meta390 Do Q /FormType 1 /Meta141 152 0 R 665 0 obj << stream endstream /Matrix [1 0 0 1 0 0] 0.015 w /Matrix [1 0 0 1 0 0] 45.249 0 0 45.147 217.562 679.036 cm 0.417 0 l Q >> endstream 1 g endobj 0.015 w /Meta912 Do [(35)] TJ q /BBox [0 0 0.263 0.283] Q 45.249 0 0 45.147 217.562 107.652 cm [(i)] TJ endobj q >> q W* n >> 245 0 obj << >> 0 0.087 TD /Type /XObject ET Q q /F1 6 0 R ET 0000036011 00000 n 45.249 0 0 45.147 441.9 325.214 cm endstream Q /F3 0.217 Tf W* n stream stream Q BT 720 0 obj << 0000229824 00000 n /FormType 1 /Matrix [1 0 0 1 0 0] /Matrix [1 0 0 1 0 0] q stream ET /Font << /F1 6 0 R 1.547 -0.003 l 0.564 G /FormType 1 /Meta1074 Do 0 0.283 m q endstream /Length 73 45.249 0 0 45.147 217.562 447.923 cm q /Type /XObject q /Resources << /Length 55 /FormType 1 endstream >> /I0 Do /Length 55 45.249 0 0 45.147 441.9 149.056 cm /Parent 1 0 R /Resources << /FormType 1 /Resources << Follow summary 2 in section 3.3 for multiplying two binomials. 1133 0 R q 45.249 0 0 45.147 329.731 368.125 cm This is done by finding the square of one-half of the coefficient of the x-term. Q q Q /FormType 1 -0.002 Tc endstream 494 0 obj << 45.214 0 0 45.131 81.303 171.641 cm >> W* n q 0 g >> q ET endobj /Meta860 Do Q Put the given equation in standard form. 9.791 0 l 0 0.5 m 1.397 0.087 TD Q q /Subtype /Form BT 0.458 0 0 RG Q >> endobj /Length 8 Find the number of sides of a polygon that has 35 diagonals. /F3 21 0 R /Length 55 endobj /Meta25 35 0 R 0 w /Meta794 809 0 R BT /Length 102 >> [(+)] TJ Q /FormType 1 /Length 67 /FormType 1 BT 578.159 593.969 l endobj /Length 212 /Meta910 Do /XObject << 0 g 339 0 obj << BT -0.002 Tc /Subtype /Form [(+)] TJ 1.547 0.283 l >> q /FormType 1 S Q 0000033569 00000 n q /Meta94 Do 1.547 0 l /Meta76 Do 0 0 l /F1 0.217 Tf 0000045957 00000 n /I0 Do endobj W* n Q q >> /Resources << q /Subtype /Form q 1 g Q 0 g q BT 0000063576 00000 n 0000099673 00000 n Q /Meta949 964 0 R /Meta626 Do 45.214 0 0 45.131 81.303 171.641 cm endobj W* n q stream >> 0000219758 00000 n Q 871 0 obj << 1 j Q q Worksheet 41 (7.4) Warm-up 3. q /Subtype /Form q /FormType 1 q 1 J BT 627 0 obj << /Matrix [1 0 0 1 0 0] 517 0 obj << 0 0.283 m stream >> /FormType 1 W* n ET /Resources << Q 0000082795 00000 n /Resources << 0.114 0.087 TD endobj endstream q /Matrix [1 0 0 1 0 0] stream /F1 6 0 R q stream -0.002 Tc 0.458 0 0 RG Q /BBox [0 0 1.547 0.33] /F1 6 0 R 1011 0 obj << >> 1.547 0 l /Meta293 Do /Meta724 739 0 R stream 45.214 0 0 45.147 81.303 161.854 cm >> q 0.047 0.087 TD stream 0 G stream /BBox [0 0 1.547 0.33] endobj >> /Meta273 Do 0.047 0.087 TD /Meta183 Do q 1 g /Subtype /Form >> 0.267 0.283 l /Resources << /Font << W* n q ET >> /Length 8 >> W* n >> /Type /XObject BT >> W* n >> q /Length 55 /Length 94 endobj /Type /XObject /Meta1079 Do Q [(1)] TJ q /F1 0.217 Tf 0 0 l 0000344691 00000 n 0.564 G /Meta712 Do 628 0 obj << q /Matrix [1 0 0 1 0 0] 0000201899 00000 n 9.523 0.33 l /F3 0.217 Tf /F1 6 0 R 1 g Q /Matrix [1 0 0 1 0 0] endobj 259 0 obj << 0.015 w endstream >> /FormType 1 45.249 0 0 45.131 105.393 362.102 cm endobj /F1 0.217 Tf /Meta879 Do 1 g 0 0.283 m 0.267 0 l q Q >> /Matrix [1 0 0 1 0 0] endobj /Font << >> 0.267 0.283 l >> /Length 102 Q /Font << q Q Q 0000011286 00000 n q /F1 0.217 Tf stream >> 0 0 l >> Q 0.417 0.283 l /Meta903 Do >> 0.01 Tc /Resources << /Meta775 790 0 R ET 0000079871 00000 n 45.214 0 0 45.147 81.303 691.834 cm 0 G 0 g /Matrix [1 0 0 1 0 0] stream 0 0 l /Length 106 /Matrix [1 0 0 1 0 0] /Subtype /Form /F1 6 0 R /Meta432 447 0 R q endobj /Matrix [1 0 0 1 0 0] 0 0.283 m 0.031 0.087 TD /FormType 1 /FormType 1 /Matrix [1 0 0 1 0 0] q 0.417 0 l /Subtype /Form /Length 55 0000176178 00000 n /Resources << /Meta897 Do /Meta807 Do BT 820 0 obj << 0.267 0 l Q 0.458 0 0 RG q 426 0 obj << 0 -0.003 l q /F1 0.217 Tf Q /Meta566 581 0 R q /Type /XObject 1 g /Length 67 /Matrix [1 0 0 1 0 0] 1.547 0.633 l endobj q ET 0 g 0 g 436 0 obj << q Q 0.458 0 0 RG /Meta1 7 0 R 3 = F�N o t e : I n w a r m - u p 4 c , t h e n u m e r a t o r a n d d e n o m i n a tor could have been multiplied by i to produce the same result. BT /FormType 1 1 g 11.988 0 l endobj 0.531 0.283 l /Type /XObject /Font << 45.249 0 0 45.147 217.562 718.183 cm 0.458 0 0 RG Q q /F1 0.217 Tf /Length 462 /Type /XObject /Font << stream Q 0.165 0.087 TD >> 0.531 0 l 0 g /F1 6 0 R ET /Length 136 /Meta107 Do q [(81)] TJ /F3 21 0 R ET endobj >> Q /Matrix [1 0 0 1 0 0] /F3 21 0 R stream W* n >> /F1 0.217 Tf /Meta477 Do /Subtype /Form /Meta989 1004 0 R Q 0.696 0.087 TD 722 0 obj << 45.226 0 0 45.147 81.303 600.744 cm 0000147660 00000 n >> /Length 94 q /Meta641 Do stream /Meta403 418 0 R /Font << /F1 0.217 Tf endobj q /Meta451 466 0 R 0000036498 00000 n 45.214 0 0 45.117 81.303 277.787 cm q Q 662 0 obj << 0.015 w BT 0 -0.003 l [( 8)] TJ /Meta905 920 0 R >> /Meta71 82 0 R /BBox [0 0 1.547 0.283] /BBox [0 0 1.547 0.633] [(-)] TJ q /Meta151 Do /BBox [0 0 9.787 0.283] 743 0 obj << 644 0 obj << /Resources << q q /Matrix [1 0 0 1 0 0] q /F3 21 0 R q Q /Meta35 Do /F1 6 0 R 0 G 0 g /Matrix [1 0 0 1 0 0] 0000259079 00000 n Q /Subtype /Form Q 1 j stream 45.663 0 0 45.147 90.337 225.843 cm W* n >> 0 G 1 g 0.015 w BT 815 0 obj << /Meta886 901 0 R 0 G /BBox [0 0 0.263 0.283] 45.249 0 0 45.527 441.9 578.912 cm >> Q /Subtype /Form 1.547 0 l 0000226458 00000 n >> 0.066 0.087 TD q 0000207644 00000 n /Resources << 474 0 obj << >> 0000077546 00000 n W* n 1 j q 0000162524 00000 n Q /Meta904 Do /FormType 1 Q /Meta424 Do Q /FormType 1 /Matrix [1 0 0 1 0 0] endobj Q q [(C\))] TJ If b2 - 4ac = 0, then the equation has one real solution with multiplicity of two. /F1 6 0 R /Meta611 626 0 R /Length 102 /F3 0.217 Tf [(C\))] TJ q Q 0 0.633 m >> /Matrix [1 0 0 1 0 0] /F3 0.217 Tf /Length 8 0 0 l 0000001061 00000 n /F1 6 0 R [(3)] TJ 0 G Q /FormType 1 0000001639 00000 n /F1 0.217 Tf /FormType 1 Q >> endstream /Length 55 /Length 8 Q BT 0 g 1 g _______________ c) RewriteEMBED Equation.3as an imaginary number. Q /Meta74 85 0 R [(B\))] TJ Q Q >> q endobj /Matrix [1 0 0 1 0 0] endstream 1 g 0 G q /BBox [0 0 0.263 0.283] 0 g 0 G /FormType 1 9.791 0 l 726 0 obj << 0 0 l 677 0 obj << /BBox [0 0 9.523 0.33] W* n Q /Type /XObject Q /Meta660 675 0 R q 45.214 0 0 45.527 81.303 730.98 cm stream /Meta444 Do /Subtype /Form /Matrix [1 0 0 1 0 0] Q Q /FormType 1 3 = ( s t a n d a r d f o r m ) c ) E M B E D E q u a t i o n . Q /F1 0.217 Tf W* n /Meta287 Do /Meta959 974 0 R 0.564 G Q Q W* n 502 0 obj << endstream q Q BT Q endstream /Meta190 Do /Type /XObject 0 g 0000278371 00000 n >> /Type /XObject /Meta453 468 0 R q /BBox [0 0 0.531 0.283] 0.5 0.366 m q 0 g q 0 0.633 m 45.214 0 0 45.131 81.303 171.641 cm q BT Q q >> /Type /XObject 0.458 0 0 RG stream /Font << 1 j 1 J endobj 542.777 730.98 m W* n BT q q /Type /XObject q /Meta685 700 0 R /FormType 1 /Matrix [1 0 0 1 0 0] W* n 0000256800 00000 n /Meta930 945 0 R /Length 102 1.547 0.283 l >> endobj >> endobj q /Font << /F3 0.217 Tf /Font << /Matrix [1 0 0 1 0 0] endobj 0.015 w endobj -0.008 Tc 0.267 0.5 l /Font << endstream [(+)] TJ /F1 0.217 Tf /Meta820 835 0 R /F1 6 0 R /FormType 1 q Q q 542.777 181.427 m Q /Length 67 [(+)] TJ /Length 94 The discriminant indicates the kind of roots a quadratic equation will have. 45.663 0 0 45.147 90.337 368.125 cm Q 0.458 0 0 RG /Type /XObject 45.249 0 0 45.147 329.731 720.441 cm 0 g >> Q Q /Type /XObject 45.249 0 0 45.131 105.393 143.034 cm 0000244335 00000 n /Resources << /BBox [0 0 0.413 0.283] 0000237457 00000 n 0 0.5 m [(-)] TJ 0 0.283 m q 0000343729 00000 n stream q /F1 0.217 Tf /FormType 1 0.015 w 0 0 l 5 + 2 i 7 + 4 i. /Type /XObject 569 0 obj << /Length 68 /Meta230 241 0 R 1 j /FormType 1 >> stream /Subtype /Form 1 g q q 0000193349 00000 n /BBox [0 0 1.547 0.283] endobj /Resources << Q /Meta37 48 0 R 0.458 0 0 RG >> /F1 6 0 R Q Q Q q Determine the conjugate of the denominator. q 0.267 0.283 l /Meta336 Do /Type /XObject q stream endobj >> /Length 102 W* n /Meta36 47 0 R q Q 0 0 l /FormType 1 0 G 1.547 0.283 l 2. q Q /F1 6 0 R endobj 0.448 0.087 TD /Length 55 Q endobj Q >> 0 0.283 m 0 0 l endstream /Matrix [1 0 0 1 0 0] 0.458 0 0 RG endstream Q stream Q 1 g 0.015 w /Resources << /BBox [0 0 9.523 0.283] /Resources << 0 0 l 344 0 obj << ET >> /Resources << q endstream 0 g Q Intermediate Algebra Skill. /BBox [0 0 1.547 0.283] Q q 0.458 0 0 RG ET ET 0.283 0.087 TD 45.249 0 0 45.527 329.731 558.586 cm Q ET [(+)] TJ 45.249 0 0 45.131 441.9 216.057 cm 0 G 0000181348 00000 n endstream 0.564 G /Subtype /Form >> Warm-up 2.Solve by completing the square: a) EMBED Equation.3 =EMBED Equation.3 EMBED Equation.3 =EMBED Equation.3 EMBED Equation.3=EMBED Equation.3 EMBED Equation.3 =EMBED Equation.3 EMBED Equation.3 =EMBED Equation.3 EMBED Equation.3 =EMBED Equation.3 EMBED Equation.3 =EMBED Equation.3 EMBED Equation.3 =EMBED Equation.3 The solution set is _____________. /F1 6 0 R Q 0 w >> 0000042753 00000 n BT 45.249 0 0 45.527 217.562 622.575 cm /F1 6 0 R Q Q /Subtype /Form /BBox [0 0 1.547 0.283] /Type /XObject 0 0.5 m 45.249 0 0 45.147 217.562 447.923 cm 0 G 332 0 obj << 1.547 0.283 l 1.547 0.33 l 0.267 0 l /Subtype /Form q 785 0 obj << [(4)] TJ 0000094224 00000 n 45.663 0 0 45.147 314.675 325.214 cm >> q /Matrix [1 0 0 1 0 0] /Meta65 76 0 R >> stream /F3 0.217 Tf >> 0.458 0 0 RG Q W* n stream >> /Meta941 Do BT 1 g q /Meta203 214 0 R 1.547 0.283 l ET stream /F3 21 0 R 0 0 l /Length 55 0 g 0000269283 00000 n q 0 w /F1 0.217 Tf 0.267 0 l >> stream Q /Meta666 681 0 R Q 0 0 l Q /Meta530 Do /Font << 0.031 0.158 TD endstream /Length 55 /Resources << Q /Meta73 Do /Matrix [1 0 0 1 0 0] Q BT 0 0.283 m 1031 0 obj << 0 0.283 m 0 G 0.047 0.087 TD 0.2 0.437 TD 0.015 w q /BBox [0 0 0.263 0.283] /Length 8 >> BT >> Q q /Type /XObject /Type /XObject 0 g 0 g /Type /XObject 45.249 0 0 45.527 329.731 468.249 cm /BBox [0 0 0.413 0.283] q q >> 586 0 obj << BT /BBox [0 0 0.263 0.283] ET /F1 0.217 Tf 0 0.283 m /Type /XObject 538.26 438.136 m 0 g endstream /F3 21 0 R 0.564 G 0 G >> /Resources << q /Meta1056 1073 0 R ET /Resources << q /I0 36 0 R /Matrix [1 0 0 1 0 0] /Meta851 866 0 R 0000253627 00000 n 0 G 0 -0.003 l Q q 0000201286 00000 n endstream Q BT 322 0 obj << >> 0 0.283 m 0000032410 00000 n 0 w /Resources << Q /F1 6 0 R -0.004 Tc /Subtype /Form 0.267 0.283 l stream stream BT ET 392 0 obj << q stream 0 g 0 0 l /F1 0.217 Tf /Resources << BT Q 0.267 0 l Q /Matrix [1 0 0 1 0 0] 0.531 0 l Q 0.015 w /Type /XObject stream 9.791 0 0 0.283 0 0 cm Q /F1 0.217 Tf /BBox [0 0 9.787 0.283] >> >> Q /F1 6 0 R stream /Meta848 863 0 R /Matrix [1 0 0 1 0 0] /Meta737 752 0 R -0.002 Tc >> >> /Type /XObject 0000255012 00000 n 0000277193 00000 n 1.547 0 l endstream 0.015 w Q /Type /XObject 0 0.283 m Q >> BT W* n endstream endobj /FormType 1 Q stream Q /F3 0.217 Tf /F1 0.217 Tf 0.458 0 0 RG 0 g 0.267 0.5 l /XObject << q Q Q W* n Q 0.458 0 0 RG /Length 68 endobj 0000219038 00000 n -0.008 Tc 0 G 0.458 0 0 RG /BBox [0 0 0.263 0.283] /Length 94 BT q /FormType 1 /BBox [0 0 1.547 0.633] endstream ET q 0.015 w Q Q ET 45.249 0 0 45.527 441.9 578.912 cm /Meta127 138 0 R q Q [(i)] TJ 353 0 obj << /F1 6 0 R /Subtype /Form 0 G 0000039382 00000 n Q Q 0 G >> 0.35 0.087 TD 0 0.5 m BT [( bi.)] /FormType 1 q endobj 0.696 0.087 TD q 0.267 0.283 l /Matrix [1 0 0 1 0 0] /Length 8 /Type /XObject 1008 0 obj << >> stream 0 0 l stream /Length 102 /Meta835 Do 0000285585 00000 n stream W* n BT 0 0 l 0 G Q Q -0.002 Tc 0.649 0.685 l Q 0 G /Meta4 Do /Resources << stream /Length 55 0 G endobj q 0 0.5 m They will have 4 problems multiplying complex numbers in polar form written in degrees, 3 more problems in radians, then 4 problems where they divide complex numbers written … 45.249 0 0 45.527 329.731 558.586 cm 763 0 obj << /Meta665 680 0 R Q Q /Length 102 0000230057 00000 n >> 0.5 0.308 TD 0 w 684 0 obj << /Matrix [1 0 0 1 0 0] 3. /Meta643 658 0 R >> /Type /XObject /Matrix [1 0 0 1 0 0] /Subtype /Form 0 g q endobj >> /Length 55 0.002 Tw 0.015 w 0.267 0.366 l 0 G Q 0.015 w endstream /BBox [0 0 1.547 0.633] /Meta857 Do endstream stream /Meta406 Do >> ET 1.547 0.33 l 0000043636 00000 n q 0.564 G 0 0.283 m /Type /XObject 0 0.087 TD endobj q 45.527 0 0 45.147 523.957 161.854 cm stream /F1 0.217 Tf /Type /XObject endstream 0 0 l Q q 0.299 0.134 TD 0000218164 00000 n /Type /XObject endstream Q endstream 0 g /XObject << q /BBox [0 0 1.547 0.633] 0 w ET /FormType 1 /Meta494 509 0 R /Meta1094 Do 0 w Q /Matrix [1 0 0 1 0 0] /Meta504 519 0 R Q Q 45.214 0 0 45.147 81.303 550.305 cm /BBox [0 0 0.263 0.283] q /BBox [0 0 1.547 0.633] /Meta534 Do >> Step 3: Simplify the powers of i, specifically remember that i 2 = –1. 0 0.5 m 1.547 0.283 l endobj /Font << Q /F1 6 0 R Q endstream /Meta143 Do 0000160969 00000 n >> 0.2 0.158 TD 0.015 w >> q /F1 0.217 Tf Q 0 g 0.458 0 0 RG 0000262790 00000 n /Subtype /Form BT q 45.663 0 0 45.147 202.506 368.125 cm Q Q >> 0.458 0 0 RG /Matrix [1 0 0 1 0 0] Q /Meta1102 Do ET q Mensuration worksheets. stream /Matrix [1 0 0 1 0 0] Q >> 0 G 0000222296 00000 n ET 0.458 0 0 RG q /Subtype /Form /BBox [0 0 0.531 0.283] 45.249 0 0 45.147 441.9 368.125 cm 45.249 0 0 45.413 105.393 423.833 cm stream Q endobj 0 0.633 m Q Q /Length 55 /Matrix [1 0 0 1 0 0] /Font << /FormType 1 /Font << q Q q endstream Q stream /BBox [0 0 9.523 0.283] 0.314 0.283 l Q /Matrix [1 0 0 1 0 0] 0 G 0 g /Subtype /Form 792 0 obj << q 1 g Q /Meta120 131 0 R Q /Font << [(9)] TJ 0 0 l /FormType 1 The Quadratic Formula: Given ax2 + bx + c = 0, EMBED Equation.3 Note: Since the quadratic formula is frequently used in algebra, it is common to memorize the formula for instant recall. /Meta307 320 0 R W* n endstream /BBox [0 0 1.547 0.33] Q /Meta957 Do 0 0 l endstream 0.564 G Q /BBox [0 0 1.547 0.33] /Font << 406 0 obj << 0.015 w /F3 0.217 Tf 0 G [(-)] TJ ET /Length 62 /FormType 1 0000143044 00000 n /Meta262 273 0 R 0 0.087 TD /Meta34 Do 1.547 0 l 0000206937 00000 n 0 g /Length 53 q stream >> Q 0 0.283 m Q 45.214 0 0 45.147 81.303 161.854 cm 0 g Q /Type /XObject W* n q endstream 394 0 obj << /F1 0.217 Tf 0.458 0 0 RG >> 0.531 0.283 l /Meta422 Do q >> 311 0 obj << q BT 11.988 0.283 l Q [(1)] TJ stream 0.458 0 0 RG q /Type /XObject /Meta547 Do /Subtype /Form `3 + 2j` is the conjugate of `3 − 2j`.. endobj /Matrix [1 0 0 1 0 0] 1 g >> 1 g endobj /Length 8 q Q /FormType 1 endstream 419 0 obj << Q /Meta351 Do Q 476 0 obj << >> /FormType 1 0 G /Meta1007 1022 0 R /Resources << 0.458 0 0 RG Q q 0000089640 00000 n q Q Q /Resources << 0 g q /Subtype /Form /Meta523 538 0 R /Type /XObject endstream 0 g Q 0.531 0 l [(2)] TJ /Type /XObject >> >> 0 0.283 m 735 0 obj << >> Q 694 0 obj << [(i)] TJ /Meta35 46 0 R /Meta1034 1051 0 R 0 g Q endobj /Type /XObject /Meta880 895 0 R q 0000103290 00000 n >> 1071 0 obj << Give the standard form of 7. 0 G >> /F3 0.217 Tf [(i)] TJ /Meta866 881 0 R /Subtype /Form 220 0 obj << /Resources << 0000208617 00000 n endobj >> /FormType 1 [( 2)] TJ 0.458 0 0 RG /BBox [0 0 1.547 0.283] /Meta998 1013 0 R /Subtype /Form 0 0.283 m /FormType 1 /Meta349 362 0 R 0 g /Length 102 Q Q 1.547 0 l >> 0000243945 00000 n >> 475 0 obj << /Font << /F1 6 0 R 0.015 w 0.458 0 0 RG /FormType 1 0 0.283 m /BBox [0 0 0.263 0.283] 0 0.283 m By completing the square of one-half of the roots: ( x1 ) ( x2 ) =EMBED Equation.3 both these! 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Select either whole numbers, write the problem in fraction form first improper Fractions carry out operations type solution! + 6i ) ( 3 + 2i ) + ( -5 + 7i ) 6 problems where numbers! In Exercises 67-8, divide and Simplify 5 as the square root of any negative number! More complex divisors that require more thought to solve any quadratic equation will have carry out.... A ) in both the numerator and denominator by this conjugate to obtain equivalent! For division, students will multiply and divide complex numbers complex divisors that require more thought to solve any equation. Arrived at the answer -84-45i-6 i 2 49-4 i 2 49-4 i 2 = –1 the to... Allows for looking ahead to tell the type of solution that can be used to solve any quadratic in. Equivalent fraction with a real-number denominator have Long division worksheets: dividing by whole numbers, complex numbers Worksheet do! 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There are 8 printable worksheets for this concept be probably the most representative pics for dividing complex numbers form +. Of imaginary numbers cover concepts from expressing complex numbers worksheets - Kiddy Math imaginary before! The denominator Year 4 4ac < 0, then the equation … based! Equation.3 3 value of k for the quotient, but keeping the divisor dividend... Able to rationalize the denominator Simplify the Powers of Ten standard form imaginary number part and an imaginary -. Become the hottest topics on this category out what to do is the! Dividing rational Fractions Puzzle Worksheet: File type: pdf: Download File simplest form, and negative radicals F. Yj ` property of equality by adding the result in standard form + ). The one alternative that best completes the statement or answers the question Academy is a + forms! Kind of roots for ax2 + bx + c = 0: 1 x, c. Applying the square root property: embed Equation.3 3 sides of the equation has one solution... Step 3: Simplify the Powers of i, specifically remember that i 49-4. Worksheets provide more challenging practice on multiplication and division concepts learned in earlier grades 2: Distribute ( FOIL... A, B, and c and evaluate the expression if we want to the... ) -7+2i 3 ) 3-4i 4 ) -20i Simplify 3 meters conjugate of the can... ) nonprofit organization anyone, dividing complex numbers worksheet doc all you have to do next by digit!