Question

find the sqroot of 7+24i
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Answer 1

Answer:

4 + 3i or -4 -3i

Step-by-step explanation:

Let the square root of 7 + 24i = a +bi where a and b are real numbers.

We know that i² = -1

Now (a+bi)² = 7 + 24i

➪ a² -b² + 2(a)(b)i =7 + 24i

Comparing both sides we get two equations a² - b² =7

And 2ab = 24

➪ab = 12

➪ b =12/a

Now putting the value of b in equation 1 we get

a² - (12/a)² = 7

➪a² -(144/a²) = 7

➪ a⁴ -144 = 7a²

➪ a⁴ - 7 a² -144 = 0

Solving above quadratic equation we get

a² = (7+√(576+49))/2 or a² = (7-√(576+49))/2

➪a² = (7+25)/2 or a² = (7–25)/2

➪a² = 16 or a² = -9

Since we have assumed a as real a² can't be negative

Therefore a² = 16

➪ a = 4 or a = -4

Putting the value of a in ab = 12 we get

b = 3 or b = -3

Therefore our required solution is 4 + 3i or -4 -3i

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☞Hope it helps