What is the answer to x^2+6x+10=i ?

The answer to x^2+6x+10=i is x=-2.54491…+1.09868…i,x=-3.45508…-1.09868…i

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x^2+6x+10=i

x^{2} + 6 x + 10 = i

x = - 2.54491 \dots + 1.09868 \dots i, x = - 3.45508 \dots - 1.09868 \dots i

Solution steps

x^{2} + 6 x + 10 = i

Substitute

x = a + bi

(a + bi)^{2} + 6 (a + bi) + 10 = i

Expand

(a + bi)^{2} + 6 (a + bi) + 10 : (a^{2} - b^{2} + 6 a + 10) + i (6 b + 2 ab)

(a^{2} - b^{2} + 6 a + 10) + i (6 b + 2 ab) = i

Rewrite

i

in standard complex form:

0 + i

(a^{2} - b^{2} + 6 a + 10) + i (6 b + 2 ab) = 0 + i

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

[a^{2} - b^{2} + 6 a + 10 = 0 6 b + 2 ab = 1]

[a^{2} - b^{2} + 6 a + 10 = 0 6 b + 2 ab = 1] : (a = - 2.54491 \dots, a = - 3.45508 \dots, b = 1.09868 \dots b = - 1.09868 \dots)

Substitute back

x = a + bi

x = - 2.54491 \dots + 1.09868 \dots i, x = - 3.45508 \dots - 1.09868 \dots i

5 x - 3 y i = 10 + 12 i

zz + 2 z = 12 + 6 i

9 + (x + 4 y) i = x + 5 i

i (x) = e x^{2} + 2 π

(1 - i) x + 3 y = 2 - 1

What is the answer to x^2+6x+10=i ?
The answer to x^2+6x+10=i is x=-2.54491…+1.09868…i,x=-3.45508…-1.09868…i