Solve |z^2 - i| = |z^2i|

Question

Solve the equation

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

Alternative Form

z_{1} \approx 0.707107 + 0.707107 i z_{2} \approx - 0.707107 - 0.707107 i

Evaluate

z^{2} - i = z^{2} i

When the expression in absolute value bars is not negative, remove the bars

z^{2} - i = z^{2} i

Simplify

More Steps

z^{2} - i = i z^{2}

Move the expression to the left side

z^{2} - i - i z^{2} = 0

Collect like terms by calculating the sum or difference of their coefficients

(1 - i) z^{2} - i = 0

Move the constant to the right-hand side and change its sign

(1 - i) z^{2} = 0 + i

Add the terms

(1 - i) z^{2} = i

Simplify

z = i

Rewrite the complex number in polar form

More Steps

z = 1 \times (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 = 1 \times (cos (\frac{\frac{π}{2} + 2 kπ}{2}) + i sin (\frac{\frac{π}{2} + 2 kπ}{2}))

Simplify

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

Solution

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

Alternative Form

z_{1} \approx 0.707107 + 0.707107 i z_{2} \approx - 0.707107 - 0.707107 i

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