Question

7-24i/3+4i find square root of it .

as soon as possible. urgent


who will wright solve it within 15 minutes , Our institution will provide his Rs2000 in cash

Answer 1

Let Z= (7 - 24i)/(3 + 4i)
multiply (3 - 4i) with numerator and denominator .

= (7 - 24i)(3 - 4i)/(3 + 4i)(3 - 4i)
= (21 - 28i - 72i +96i²)/(3² - 4²i²)
= ( 21 - 96 - 100i)/(9 + 16) [ Because i² = -1 ]
= (- 75 - 100i)/(25)
= -3 - 4i


now , Z = -3 - 4i
we have to find square root of Z = -3 - 4i
Let √{-3-4i}= a + ib
it means , -3 - 4i = (a + ib)²
-3 - 4i = (a² - b²) + 2iab
now, compare both sides,
-3 = (a² - b²) and 2ab = -4 => ab = -2

-3 = a² - (-2)²/a²
-3a² = a⁴ - 4
a⁴ + 3a² - 4 = 0
(a² + 4)(a² - 1) =0
so, a = ±1 or ±2i
Then, b = -+2 or -+1

Hence, a+ib = ±( 1 - 2i)