Solve x^2=22x+10

$$x ^ { 2 } = 22 x + 10$$

Solve for x

$$x=\sqrt{131}+11\approx 22.445523142$$

$$x=11-\sqrt{131}\approx -0.445523142$$

Steps Using the Quadratic Formula

Subtract 22x from both sides.

$$x^{2}-22x=10$$

Subtract 10 from both sides.

$$x^{2}-22x-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -22 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\left(-10\right)}}{2}$$

Square -22.

$$x=\frac{-\left(-22\right)±\sqrt{484-4\left(-10\right)}}{2}$$

Multiply -4 times -10.

$$x=\frac{-\left(-22\right)±\sqrt{484+40}}{2}$$

Add 484 to 40.

$$x=\frac{-\left(-22\right)±\sqrt{524}}{2}$$

Take the square root of 524.

$$x=\frac{-\left(-22\right)±2\sqrt{131}}{2}$$

The opposite of -22 is 22.

$$x=\frac{22±2\sqrt{131}}{2}$$

Now solve the equation x=\frac{22±2\sqrt{131}}{2} when ± is plus. Add 22 to 2\sqrt{131}.

$$x=\frac{2\sqrt{131}+22}{2}$$

Divide 22+2\sqrt{131} by 2.

$$x=\sqrt{131}+11$$

Now solve the equation x=\frac{22±2\sqrt{131}}{2} when ± is minus. Subtract 2\sqrt{131} from 22.

$$x=\frac{22-2\sqrt{131}}{2}$$

Divide 22-2\sqrt{131} by 2.

$$x=11-\sqrt{131}$$

The equation is now solved.

$$x=\sqrt{131}+11 x=11-\sqrt{131}$$

Steps for Completing the Square

Subtract 22x from both sides.

$$x^{2}-22x=10$$

Divide -22, the coefficient of the x term, by 2 to get -11. Then add the square of -11 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-22x+\left(-11\right)^{2}=10+\left(-11\right)^{2}$$

Square -11.

$$x^{2}-22x+121=10+121$$

Add 10 to 121.

$$x^{2}-22x+121=131$$

Factor x^{2}-22x+121. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-11\right)^{2}=131$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-11\right)^{2}}=\sqrt{131}$$

Simplify.

$$x-11=\sqrt{131} x-11=-\sqrt{131}$$

Add 11 to both sides of the equation.

$$x=\sqrt{131}+11 x=11-\sqrt{131}$$