Solve x^2=22x-5

$$x ^ { 2 } = 22 x - 5$$

Solve for x

$$x=2\sqrt{29}+11\approx 21.770329614$$

$$x=11-2\sqrt{29}\approx 0.229670386$$

Steps Using the Quadratic Formula

Subtract 22x from both sides.

$$x^{2}-22x=-5$$

Add 5 to both sides.

$$x^{2}-22x+5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -22 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\times 5}}{2}$$

Square -22.

$$x=\frac{-\left(-22\right)±\sqrt{484-4\times 5}}{2}$$

Multiply -4 times 5.

$$x=\frac{-\left(-22\right)±\sqrt{484-20}}{2}$$

Add 484 to -20.

$$x=\frac{-\left(-22\right)±\sqrt{464}}{2}$$

Take the square root of 464.

$$x=\frac{-\left(-22\right)±4\sqrt{29}}{2}$$

The opposite of -22 is 22.

$$x=\frac{22±4\sqrt{29}}{2}$$

Now solve the equation x=\frac{22±4\sqrt{29}}{2} when ± is plus. Add 22 to 4\sqrt{29}.

$$x=\frac{4\sqrt{29}+22}{2}$$

Divide 22+4\sqrt{29} by 2.

$$x=2\sqrt{29}+11$$

Now solve the equation x=\frac{22±4\sqrt{29}}{2} when ± is minus. Subtract 4\sqrt{29} from 22.

$$x=\frac{22-4\sqrt{29}}{2}$$

Divide 22-4\sqrt{29} by 2.

$$x=11-2\sqrt{29}$$

The equation is now solved.

$$x=2\sqrt{29}+11 x=11-2\sqrt{29}$$

Steps for Completing the Square

Subtract 22x from both sides.

$$x^{2}-22x=-5$$

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.

$$x^{2}-22x+\left(-11\right)^{2}=-5+\left(-11\right)^{2}$$

Square -11.

$$x^{2}-22x+121=-5+121$$

Add -5 to 121.

$$x^{2}-22x+121=116$$

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}.

$$\left(x-11\right)^{2}=116$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-11\right)^{2}}=\sqrt{116}$$

Simplify.

$$x-11=2\sqrt{29} x-11=-2\sqrt{29}$$

Add 11 to both sides of the equation.

$$x=2\sqrt{29}+11 x=11-2\sqrt{29}$$