Solve z^2=5i - ScanMath

Question

Solve the equation

z_{1} = \frac{10}{2} + \frac{10}{2} i z_{2} = - \frac{10}{2} - \frac{10}{2} i

Alternative Form

z_{1} \approx 1.581139 + 1.581139 i z_{2} \approx - 1.581139 - 1.581139 i

Evaluate

z^{2} = 5 i

Simplify

z = 5 i

Rewrite the complex number in polar form

More Steps

z = 5 (cos (\frac{π}{2}) + i sin (\frac{π}{2}))

Calculate the nth roots of a complex r (cos (θ) + i \times sin (θ),using n z = n r (cos \frac{θ + 2 kπ}{n} + i sin \frac{θ + 2 kπ}{n})

z = 5 \times (cos (\frac{\frac{π}{2} + 2 kπ}{2}) + i sin (\frac{\frac{π}{2} + 2 kπ}{2}))

Since n = 2,substitute k = 0, 1 into the expression

z_{1} = 5 \times (cos (\frac{\frac{π}{2} + 2 \times 0 \times π}{2}) + i sin (\frac{\frac{π}{2} + 2 \times 0 \times π}{2})) z_{2} = 5 \times (cos (\frac{\frac{π}{2} + 2 \times 1 \times π}{2}) + i sin (\frac{\frac{π}{2} + 2 \times 1 \times π}{2}))

Calculate

More Steps

z_{1} = 5 \times (cos (\frac{π}{4}) + i sin (\frac{π}{4})) z_{2} = 5 \times (cos (\frac{\frac{π}{2} + 2 \times 1 \times π}{2}) + i sin (\frac{\frac{π}{2} + 2 \times 1 \times π}{2}))

Calculate

More Steps

z_{1} = 5 \times (cos (\frac{π}{4}) + i sin (\frac{π}{4})) z_{2} = 5 \times (cos (\frac{5 π}{4}) + i sin (\frac{5 π}{4}))

Solution

z_{1} = \frac{10}{2} + \frac{10}{2} i z_{2} = - \frac{10}{2} - \frac{10}{2} i

Alternative Form

z_{1} \approx 1.581139 + 1.581139 i z_{2} \approx - 1.581139 - 1.581139 i

Show Solution