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Find the complex numbers z such that z” =5+12i You need to find the complex 7 TW number such that z?=5 + 127 Let z= a+ biwhere a,b ER, then (a+bi) =5+12i 2 22 a + 2a +b*? 254121 Hinata mero ») (a°—b*)+2abi=5412i ———— Real: a°-b° =5 Saal Equate real parts and equate Imaginary: 2ab=12 = a= mS | wl at ata CY ees ——___@) iE a5 | | Substitute for aand solve . Fi simultaneously. 36—b" =5p' b+ +5b°-36=0 « L| Solve the quadratic in b? b? =4 or-9 > b=42 bis a real number so b?= 4 6 ; aap ans => 2=3+2iorz=-3-2) e————————_ Give both possible answers. - S Exercise 1.1B Reasoning and problem-solving 1 Find the real numbers a and b such that 9 Calculate the square roots of 2—4V2i (2a+i) =35—bi 2 Find the values a, be R that satisfy the equation (2—bi)(3+ 4i) = a-13i 10 Solve these simultaneous equations, where wand z are complex numbers. 3 Solve the equation wi +z =0 2(2—5i)=13-18i for zeC 2-3w=10 4 Solve the equation 11 Solve each of these equations to find ze C (x+3i)(LO—7i)=34+6i a 2?-22=-50 bo 2*(14-)=7-171 fora complex number x 12 Simplify each of these expressions, giving 5 Find the complex numbers z, and z, such your answer in the form a+ bi £ that z, +z, =11-3i and z,—z,=5+7i in each case. > 6 Find the complex numbers z and w that a (142i)! b (2-si)° ® satisfy the equations — Oo 2+2w=6i and 32—4w =20+231 celal Us) 7 Find z,,z, €Csuch that 13 Find a, b €R such that 1 2z,-3z, =10+8i and 5z, ——z, =4+6i (a+i)' =28+bi 8 Find the complex numbers w in each of these cases. a ow=30i-16 bow =-3-4i © w*-1=20(1-i) 
