Solve 76x-76=x^2+8x

$$76 x - 76 = x ^ { 2 } + 8 x$$

Solve for x

$$x=6\sqrt{30}+34\approx 66.86335345$$

$$x=34-6\sqrt{30}\approx 1.13664655$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

$$76x-76-x^{2}=8x$$

Subtract 8x from both sides.

$$76x-76-x^{2}-8x=0$$

Combine 76x and -8x to get 68x.

$$68x-76-x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-x^{2}+68x-76=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 68 for b, and -76 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-68±\sqrt{68^{2}-4\left(-1\right)\left(-76\right)}}{2\left(-1\right)}$$

Square 68.

$$x=\frac{-68±\sqrt{4624-4\left(-1\right)\left(-76\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-68±\sqrt{4624+4\left(-76\right)}}{2\left(-1\right)}$$

Multiply 4 times -76.

$$x=\frac{-68±\sqrt{4624-304}}{2\left(-1\right)}$$

Add 4624 to -304.

$$x=\frac{-68±\sqrt{4320}}{2\left(-1\right)}$$

Take the square root of 4320.

$$x=\frac{-68±12\sqrt{30}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-68±12\sqrt{30}}{-2}$$

Now solve the equation x=\frac{-68±12\sqrt{30}}{-2} when ± is plus. Add -68 to 12\sqrt{30}.

$$x=\frac{12\sqrt{30}-68}{-2}$$

Divide -68+12\sqrt{30} by -2.

$$x=34-6\sqrt{30}$$

Now solve the equation x=\frac{-68±12\sqrt{30}}{-2} when ± is minus. Subtract 12\sqrt{30} from -68.

$$x=\frac{-12\sqrt{30}-68}{-2}$$

Divide -68-12\sqrt{30} by -2.

$$x=6\sqrt{30}+34$$

The equation is now solved.

$$x=34-6\sqrt{30} x=6\sqrt{30}+34$$

Steps for Completing the Square

Subtract x^{2} from both sides.

$$76x-76-x^{2}=8x$$

Subtract 8x from both sides.

$$76x-76-x^{2}-8x=0$$

Combine 76x and -8x to get 68x.

$$68x-76-x^{2}=0$$

Add 76 to both sides. Anything plus zero gives itself.

$$68x-x^{2}=76$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-x^{2}+68x=76$$

Divide both sides by -1.

$$\frac{-x^{2}+68x}{-1}=\frac{76}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{68}{-1}x=\frac{76}{-1}$$

Divide 68 by -1.

$$x^{2}-68x=\frac{76}{-1}$$

Divide 76 by -1.

$$x^{2}-68x=-76$$

Divide -68, the coefficient of the x term, by 2 to get -34. Then add the square of -34 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-68x+\left(-34\right)^{2}=-76+\left(-34\right)^{2}$$

Square -34.

$$x^{2}-68x+1156=-76+1156$$

Add -76 to 1156.

$$x^{2}-68x+1156=1080$$

Factor x^{2}-68x+1156. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-34\right)^{2}=1080$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-34\right)^{2}}=\sqrt{1080}$$

Simplify.

$$x-34=6\sqrt{30} x-34=-6\sqrt{30}$$

Add 34 to both sides of the equation.

$$x=6\sqrt{30}+34 x=34-6\sqrt{30}$$