Solve x^2-4x+16=121+16

$$x ^ { 2 } - 4 x + 16 = 121 + 16$$

Solve for x

$$x=5\sqrt{5}+2\approx 13.180339887$$

$$x=2-5\sqrt{5}\approx -9.180339887$$

Steps Using the Quadratic Formula

Add 121 and 16 to get 137.

$$x^{2}-4x+16=137$$

Subtract 137 from both sides.

$$x^{2}-4x+16-137=0$$

Subtract 137 from 16 to get -121.

$$x^{2}-4x-121=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and -121 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-121\right)}}{2}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-121\right)}}{2}$$

Multiply -4 times -121.

$$x=\frac{-\left(-4\right)±\sqrt{16+484}}{2}$$

Add 16 to 484.

$$x=\frac{-\left(-4\right)±\sqrt{500}}{2}$$

Take the square root of 500.

$$x=\frac{-\left(-4\right)±10\sqrt{5}}{2}$$

The opposite of -4 is 4.

$$x=\frac{4±10\sqrt{5}}{2}$$

Now solve the equation x=\frac{4±10\sqrt{5}}{2} when ± is plus. Add 4 to 10\sqrt{5}.

$$x=\frac{10\sqrt{5}+4}{2}$$

Divide 4+10\sqrt{5} by 2.

$$x=5\sqrt{5}+2$$

Now solve the equation x=\frac{4±10\sqrt{5}}{2} when ± is minus. Subtract 10\sqrt{5} from 4.

$$x=\frac{4-10\sqrt{5}}{2}$$

Divide 4-10\sqrt{5} by 2.

$$x=2-5\sqrt{5}$$

The equation is now solved.

$$x=5\sqrt{5}+2 x=2-5\sqrt{5}$$

Steps for Completing the Square

Add 121 and 16 to get 137.

$$x^{2}-4x+16=137$$

Subtract 16 from both sides.

$$x^{2}-4x=137-16$$

Subtract 16 from 137 to get 121.

$$x^{2}-4x=121$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-4x+\left(-2\right)^{2}=121+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=121+4$$

Add 121 to 4.

$$x^{2}-4x+4=125$$

Factor x^{2}-4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-2\right)^{2}=125$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{125}$$

Simplify.

$$x-2=5\sqrt{5} x-2=-5\sqrt{5}$$

Add 2 to both sides of the equation.

$$x=5\sqrt{5}+2 x=2-5\sqrt{5}$$