What is the answer to x^2-x-i=0 ?

The answer to x^2-x-i=0 is x=-0.30024…-0.62481…i,x=1.30024…+0.62481…i

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x^2-x-i=0

x^{2} - x - i = 0

x = - 0.30024 \dots - 0.62481 \dots i, x = 1.30024 \dots + 0.62481 \dots i

Solution steps

x^{2} - x - i = 0

Substitute

x = a + bi

(a + bi)^{2} - (a + bi) - i = 0

Expand

(a + bi)^{2} - (a + bi) - i : (a^{2} - b^{2} - a) + i (- 1 - b + 2 ab)

(a^{2} - b^{2} - a) + i (- 1 - b + 2 ab) = 0

Rewrite

0

in standard complex form:

0 + 0 i

(a^{2} - b^{2} - a) + i (- 1 - b + 2 ab) = 0 + 0 i

Complex numbers can be equal only if their real and imaginary parts are equalRewrite as system of equations:

[a^{2} - b^{2} - a = 0 - 1 - b + 2 ab = 0]

[a^{2} - b^{2} - a = 0 - 1 - b + 2 ab = 0] : (a = - 0.30024 \dots, a = 1.30024 \dots, b = - 0.62481 \dots b = 0.62481 \dots)

Substitute back

x = a + bi

x = - 0.30024 \dots - 0.62481 \dots i, x = 1.30024 \dots + 0.62481 \dots i

∣ z ∣ = 2 - 3 i + z

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

(2 + i) x = y

3 x - 4 y + (6 x + 2 y) i = 5 i

s i \frac{x}{3} + 2 x = 7, x

What is the answer to x^2-x-i=0 ?
The answer to x^2-x-i=0 is x=-0.30024…-0.62481…i,x=1.30024…+0.62481…i