Solve 442=x^2+2x

$$442= { x }^{ 2 } +2x$$

Solve for x (complex solution)

$$x=\sqrt{443}-1\approx 20.04756518$$

$$x=-\left(\sqrt{443}+1\right)\approx -22.04756518$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+2x=442$$

Subtract 442 from both sides.

$$x^{2}+2x-442=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -442 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-442\right)}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-442\right)}}{2}$$

Multiply -4 times -442.

$$x=\frac{-2±\sqrt{4+1768}}{2}$$

Add 4 to 1768.

$$x=\frac{-2±\sqrt{1772}}{2}$$

Take the square root of 1772.

$$x=\frac{-2±2\sqrt{443}}{2}$$

Now solve the equation x=\frac{-2±2\sqrt{443}}{2} when ± is plus. Add -2 to 2\sqrt{443}.

$$x=\frac{2\sqrt{443}-2}{2}$$

Divide -2+2\sqrt{443} by 2.

$$x=\sqrt{443}-1$$

Now solve the equation x=\frac{-2±2\sqrt{443}}{2} when ± is minus. Subtract 2\sqrt{443} from -2.

$$x=\frac{-2\sqrt{443}-2}{2}$$

Divide -2-2\sqrt{443} by 2.

$$x=-\sqrt{443}-1$$

The equation is now solved.

$$x=\sqrt{443}-1 x=-\sqrt{443}-1$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+2x=442$$

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.

$$x^{2}+2x+1^{2}=442+1^{2}$$

Square 1.

$$x^{2}+2x+1=442+1$$

Add 442 to 1.

$$x^{2}+2x+1=443$$

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}.

$$\left(x+1\right)^{2}=443$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{443}$$

Simplify.

$$x+1=\sqrt{443} x+1=-\sqrt{443}$$

Subtract 1 from both sides of the equation.

$$x=\sqrt{443}-1 x=-\sqrt{443}-1$$

Solve for x

$$x=\sqrt{443}-1\approx 20.04756518$$

$$x=-\sqrt{443}-1\approx -22.04756518$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+2x=442$$

Subtract 442 from both sides.

$$x^{2}+2x-442=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -442 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-442\right)}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-442\right)}}{2}$$

Multiply -4 times -442.

$$x=\frac{-2±\sqrt{4+1768}}{2}$$

Add 4 to 1768.

$$x=\frac{-2±\sqrt{1772}}{2}$$

Take the square root of 1772.

$$x=\frac{-2±2\sqrt{443}}{2}$$

Now solve the equation x=\frac{-2±2\sqrt{443}}{2} when ± is plus. Add -2 to 2\sqrt{443}.

$$x=\frac{2\sqrt{443}-2}{2}$$

Divide -2+2\sqrt{443} by 2.

$$x=\sqrt{443}-1$$

Now solve the equation x=\frac{-2±2\sqrt{443}}{2} when ± is minus. Subtract 2\sqrt{443} from -2.

$$x=\frac{-2\sqrt{443}-2}{2}$$

Divide -2-2\sqrt{443} by 2.

$$x=-\sqrt{443}-1$$

The equation is now solved.

$$x=\sqrt{443}-1 x=-\sqrt{443}-1$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+2x=442$$

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.

$$x^{2}+2x+1^{2}=442+1^{2}$$

Square 1.

$$x^{2}+2x+1=442+1$$

Add 442 to 1.

$$x^{2}+2x+1=443$$

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}.

$$\left(x+1\right)^{2}=443$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{443}$$

Simplify.

$$x+1=\sqrt{443} x+1=-\sqrt{443}$$

Subtract 1 from both sides of the equation.

$$x=\sqrt{443}-1 x=-\sqrt{443}-1$$