Solve 3^2+x^2=2x-3^2

$${ 3 }^{ 2 } + { x }^{ 2 } = 2x- { 3 }^{ 2 }$$

Solve for x (complex solution)

$$x=1+\sqrt{17}i\approx 1+4.123105626i$$

$$x=-\sqrt{17}i+1\approx 1-4.123105626i$$

Steps Using the Quadratic Formula

Calculate 3 to the power of 2 and get 9.

$$9+x^{2}=2x-3^{2}$$

Calculate 3 to the power of 2 and get 9.

$$9+x^{2}=2x-9$$

Subtract 2x from both sides.

$$9+x^{2}-2x=-9$$

Add 9 to both sides.

$$9+x^{2}-2x+9=0$$

Add 9 and 9 to get 18.

$$18+x^{2}-2x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}-2x+18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and 18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 18}}{2}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 18}}{2}$$

Multiply -4 times 18.

$$x=\frac{-\left(-2\right)±\sqrt{4-72}}{2}$$

Add 4 to -72.

$$x=\frac{-\left(-2\right)±\sqrt{-68}}{2}$$

Take the square root of -68.

$$x=\frac{-\left(-2\right)±2\sqrt{17}i}{2}$$

The opposite of -2 is 2.

$$x=\frac{2±2\sqrt{17}i}{2}$$

Now solve the equation x=\frac{2±2\sqrt{17}i}{2} when ± is plus. Add 2 to 2i\sqrt{17}.

$$x=\frac{2+2\sqrt{17}i}{2}$$

Divide 2+2i\sqrt{17} by 2.

$$x=1+\sqrt{17}i$$

Now solve the equation x=\frac{2±2\sqrt{17}i}{2} when ± is minus. Subtract 2i\sqrt{17} from 2.

$$x=\frac{-2\sqrt{17}i+2}{2}$$

Divide 2-2i\sqrt{17} by 2.

$$x=-\sqrt{17}i+1$$

The equation is now solved.

$$x=1+\sqrt{17}i x=-\sqrt{17}i+1$$

Steps for Completing the Square

Calculate 3 to the power of 2 and get 9.

$$9+x^{2}=2x-3^{2}$$

Calculate 3 to the power of 2 and get 9.

$$9+x^{2}=2x-9$$

Subtract 2x from both sides.

$$9+x^{2}-2x=-9$$

Subtract 9 from both sides.

$$x^{2}-2x=-9-9$$

Subtract 9 from -9 to get -18.

$$x^{2}-2x=-18$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=-18+1$$

Add -18 to 1.

$$x^{2}-2x+1=-17$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=-17$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{-17}$$

Simplify.

$$x-1=\sqrt{17}i x-1=-\sqrt{17}i$$

Add 1 to both sides of the equation.

$$x=1+\sqrt{17}i x=-\sqrt{17}i+1$$