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प्रश्न
Find the square roots of 4 + 3i
उत्तर
|4 + 3i| = `sqrt(4^2 + 3^2)`
= `sqrt(16 + 9)`
`sqrt(25)` = 5
Let `sqrt(4 + 3"i")` = a + ib
Squaring on both sides
4 + 3i = (a + ib)2
4 + 3i = (a2 – b2) + 2 jab
Equating real and imaginary parts
a2 – b2 = 4
2ab = 3
(a2 + b2)2 = (a2 – b2)2 + 4a2 b2
= (4)2 + (3)2
= 16 + 9 = 25
∴ a2 + b2 = 5
Solving a2 – b2 = 4 and a2 + b2 = 5.
We get a2 = `9/2`
b² = `1/2`
a = `+- 3/sqrt(2)` and b = `+- 1/sqrt(2)`
∴ `sqrt(4 + 3"i")` = a + ib
= `+- (3/sqrt(2) + +- "i"/sqrt(2))`
Aliter:
Square root of 4 + 3i
Formula method
`sqrt("a" + "ib") = +- [sqrt((sqrt("a"^2 + "b"^2) + "a")/2) + "i" "b"/|"b"| sqrt((sqrt("a"^2 + "b"^2) - "a")/2)]`
Now |4 + 3i| = `sqrt(4^2 + 3^2)`
= `sqrt(16 + 9)`
= `sqrt(25)`
= 5
∴ `sqrt(4 + 3"i") = +- [sqrt((5 + 4)/2) + "i" sqrt((5 - 4)/2)]`
= `+- [3/sqrt(2) + "i" 1/sqrt(2)]`
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