Solve x^2-7x-30=x

$$x ^ { 2 } - 7 x - 30 = x$$

Solve for x

$$x=\sqrt{46}+4\approx 10.782329983$$

$$x=4-\sqrt{46}\approx -2.782329983$$

Steps Using the Quadratic Formula

Subtract x from both sides.

$$x^{2}-7x-30-x=0$$

Combine -7x and -x to get -8x.

$$x^{2}-8x-30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -8 for b, and -30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-30\right)}}{2}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-30\right)}}{2}$$

Multiply -4 times -30.

$$x=\frac{-\left(-8\right)±\sqrt{64+120}}{2}$$

Add 64 to 120.

$$x=\frac{-\left(-8\right)±\sqrt{184}}{2}$$

Take the square root of 184.

$$x=\frac{-\left(-8\right)±2\sqrt{46}}{2}$$

The opposite of -8 is 8.

$$x=\frac{8±2\sqrt{46}}{2}$$

Now solve the equation x=\frac{8±2\sqrt{46}}{2} when ± is plus. Add 8 to 2\sqrt{46}.

$$x=\frac{2\sqrt{46}+8}{2}$$

Divide 8+2\sqrt{46} by 2.

$$x=\sqrt{46}+4$$

Now solve the equation x=\frac{8±2\sqrt{46}}{2} when ± is minus. Subtract 2\sqrt{46} from 8.

$$x=\frac{8-2\sqrt{46}}{2}$$

Divide 8-2\sqrt{46} by 2.

$$x=4-\sqrt{46}$$

The equation is now solved.

$$x=\sqrt{46}+4 x=4-\sqrt{46}$$

Steps for Completing the Square

Subtract x from both sides.

$$x^{2}-7x-30-x=0$$

Combine -7x and -x to get -8x.

$$x^{2}-8x-30=0$$

Add 30 to both sides. Anything plus zero gives itself.

$$x^{2}-8x=30$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-8x+\left(-4\right)^{2}=30+\left(-4\right)^{2}$$

Square -4.

$$x^{2}-8x+16=30+16$$

Add 30 to 16.

$$x^{2}-8x+16=46$$

Factor x^{2}-8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-4\right)^{2}=46$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{46}$$

Simplify.

$$x-4=\sqrt{46} x-4=-\sqrt{46}$$

Add 4 to both sides of the equation.

$$x=\sqrt{46}+4 x=4-\sqrt{46}$$