Solve x^2-32x+600=0

$${ x }^{ 2 } -32x+600=0$$

Solve for x (complex solution)

$$x=16+2\sqrt{86}i\approx 16+18.547236991i$$

$$x=-2\sqrt{86}i+16\approx 16-18.547236991i$$

Steps Using the Quadratic Formula

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.

$$x^{2}-32x+600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -32 for b, and 600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\times 600}}{2}$$

Square -32.

$$x=\frac{-\left(-32\right)±\sqrt{1024-4\times 600}}{2}$$

Multiply -4 times 600.

$$x=\frac{-\left(-32\right)±\sqrt{1024-2400}}{2}$$

Add 1024 to -2400.

$$x=\frac{-\left(-32\right)±\sqrt{-1376}}{2}$$

Take the square root of -1376.

$$x=\frac{-\left(-32\right)±4\sqrt{86}i}{2}$$

The opposite of -32 is 32.

$$x=\frac{32±4\sqrt{86}i}{2}$$

Now solve the equation x=\frac{32±4\sqrt{86}i}{2} when ± is plus. Add 32 to 4i\sqrt{86}.

$$x=\frac{32+4\sqrt{86}i}{2}$$

Divide 32+4i\sqrt{86} by 2.

$$x=16+2\sqrt{86}i$$

Now solve the equation x=\frac{32±4\sqrt{86}i}{2} when ± is minus. Subtract 4i\sqrt{86} from 32.

$$x=\frac{-4\sqrt{86}i+32}{2}$$

Divide 32-4i\sqrt{86} by 2.

$$x=-2\sqrt{86}i+16$$

The equation is now solved.

$$x=16+2\sqrt{86}i x=-2\sqrt{86}i+16$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}-32x+600=0$$

Subtract 600 from both sides of the equation.

$$x^{2}-32x+600-600=-600$$

Subtracting 600 from itself leaves 0.

$$x^{2}-32x=-600$$

Divide -32, the coefficient of the x term, by 2 to get -16. Then add the square of -16 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-32x+\left(-16\right)^{2}=-600+\left(-16\right)^{2}$$

Square -16.

$$x^{2}-32x+256=-600+256$$

Add -600 to 256.

$$x^{2}-32x+256=-344$$

Factor x^{2}-32x+256. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-16\right)^{2}=-344$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-16\right)^{2}}=\sqrt{-344}$$

Simplify.

$$x-16=2\sqrt{86}i x-16=-2\sqrt{86}i$$

Add 16 to both sides of the equation.

$$x=16+2\sqrt{86}i x=-2\sqrt{86}i+16$$