Solve x^2-7x-6=2x+2

$${ x }^{ 2 } -7x-6=2x+2$$

Solve for x

$$x = \frac{\sqrt{113} + 9}{2} \approx 9.815072906$$

$$x=\frac{9-\sqrt{113}}{2}\approx -0.815072906$$

Steps Using the Quadratic Formula

Subtract 2x from both sides.

$$x^{2}-7x-6-2x=2$$

Combine -7x and -2x to get -9x.

$$x^{2}-9x-6=2$$

Subtract 2 from both sides.

$$x^{2}-9x-6-2=0$$

Subtract 2 from -6 to get -8.

$$x^{2}-9x-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -9 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-8\right)}}{2}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\left(-8\right)}}{2}$$

Multiply -4 times -8.

$$x=\frac{-\left(-9\right)±\sqrt{81+32}}{2}$$

Add 81 to 32.

$$x=\frac{-\left(-9\right)±\sqrt{113}}{2}$$

The opposite of -9 is 9.

$$x=\frac{9±\sqrt{113}}{2}$$

Now solve the equation x=\frac{9±\sqrt{113}}{2} when ± is plus. Add 9 to \sqrt{113}.

$$x=\frac{\sqrt{113}+9}{2}$$

Now solve the equation x=\frac{9±\sqrt{113}}{2} when ± is minus. Subtract \sqrt{113} from 9.

$$x=\frac{9-\sqrt{113}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{113}+9}{2} x=\frac{9-\sqrt{113}}{2}$$

Steps for Completing the Square

Subtract 2x from both sides.

$$x^{2}-7x-6-2x=2$$

Combine -7x and -2x to get -9x.

$$x^{2}-9x-6=2$$

Add 6 to both sides.

$$x^{2}-9x=2+6$$

Add 2 and 6 to get 8.

$$x^{2}-9x=8$$

Divide -9, the coefficient of the x term, by 2 to get -\frac{9}{2}. Then add the square of -\frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=8+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-9x+\frac{81}{4}=8+\frac{81}{4}$$

Add 8 to \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=\frac{113}{4}$$

Factor x^{2}-9x+\frac{81}{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}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{113}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{113}{4}}$$

Simplify.

$$x-\frac{9}{2}=\frac{\sqrt{113}}{2} x-\frac{9}{2}=-\frac{\sqrt{113}}{2}$$

Add \frac{9}{2} to both sides of the equation.

$$x=\frac{\sqrt{113}+9}{2} x=\frac{9-\sqrt{113}}{2}$$