What is the answer to i(x)=x^2+4x+7 ?

The answer to i(x)=x^2+4x+7 is x=-1.46798…-1.37963…i,x=-2.53201…+2.37963…i

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i(x)=x^2+4x+7

i (x) = x^{2} + 4 x + 7

x = - 1.46798 \dots - 1.37963 \dots i, x = - 2.53201 \dots + 2.37963 \dots i

Solution steps

i (x) = x^{2} + 4 x + 7

Substitute

x = a + bi

i (a + bi) = (a + bi)^{2} + 4 (a + bi) + 7

Expand

- b + ia = (a^{2} - b^{2} + 4 a + 7) + i (4 b + 2 ab)

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

[- b = a^{2} - b^{2} + 4 a + 7 a = 4 b + 2 ab]

[- b = a^{2} - b^{2} + 4 a + 7 a = 4 b + 2 ab] : (a = - 1.46798 \dots, a = - 2.53201 \dots, b = - 1.37963 \dots b = 2.37963 \dots)

Substitute back

x = a + bi

x = - 1.46798 \dots - 1.37963 \dots i, x = - 2.53201 \dots + 2.37963 \dots i

z = 2 e^{i (\frac{π}{3})}

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

z^{2} = 2 + 2 i 3

x^{2} - 3 i x - 2 = 0

6 + (x + 3 y) i = x + 3 i

What is the answer to i(x)=x^2+4x+7 ?
The answer to i(x)=x^2+4x+7 is x=-1.46798…-1.37963…i,x=-2.53201…+2.37963…i