Solve 0=x^2-16x-126

$$0 = { x }^{ 2 } -16x-126$$

Solve for x

$$x=\sqrt{190}+8\approx 21.784048752$$

$$x=8-\sqrt{190}\approx -5.784048752$$

Steps Using the Quadratic Formula

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

$$x^{2}-16x-126=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -16 for b, and -126 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\left(-126\right)}}{2}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\left(-126\right)}}{2}$$

Multiply -4 times -126.

$$x=\frac{-\left(-16\right)±\sqrt{256+504}}{2}$$

Add 256 to 504.

$$x=\frac{-\left(-16\right)±\sqrt{760}}{2}$$

Take the square root of 760.

$$x=\frac{-\left(-16\right)±2\sqrt{190}}{2}$$

The opposite of -16 is 16.

$$x=\frac{16±2\sqrt{190}}{2}$$

Now solve the equation x=\frac{16±2\sqrt{190}}{2} when ± is plus. Add 16 to 2\sqrt{190}.

$$x=\frac{2\sqrt{190}+16}{2}$$

Divide 16+2\sqrt{190} by 2.

$$x=\sqrt{190}+8$$

Now solve the equation x=\frac{16±2\sqrt{190}}{2} when ± is minus. Subtract 2\sqrt{190} from 16.

$$x=\frac{16-2\sqrt{190}}{2}$$

Divide 16-2\sqrt{190} by 2.

$$x=8-\sqrt{190}$$

The equation is now solved.

$$x=\sqrt{190}+8 x=8-\sqrt{190}$$

Steps for Completing the Square

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

$$x^{2}-16x-126=0$$

Add 126 to both sides. Anything plus zero gives itself.

$$x^{2}-16x=126$$

Divide -16, the coefficient of the x term, by 2 to get -8. Then add the square of -8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-16x+\left(-8\right)^{2}=126+\left(-8\right)^{2}$$

Square -8.

$$x^{2}-16x+64=126+64$$

Add 126 to 64.

$$x^{2}-16x+64=190$$

Factor x^{2}-16x+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-8\right)^{2}=190$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-8\right)^{2}}=\sqrt{190}$$

Simplify.

$$x-8=\sqrt{190} x-8=-\sqrt{190}$$

Add 8 to both sides of the equation.

$$x=\sqrt{190}+8 x=8-\sqrt{190}$$