Solve 3(x+2)(x-2)=(x-4)+8x

$$3(x+2)(x-2)=(x-4)+8x$$

Solve for x

$$x=\frac{\sqrt{177}}{6}+\frac{3}{2}\approx 3.717355783$$

$$x=-\frac{\sqrt{177}}{6}+\frac{3}{2}\approx -0.717355783$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3 by x+2.

$$\left(3x+6\right)\left(x-2\right)=x-4+8x$$

Use the distributive property to multiply 3x+6 by x-2 and combine like terms.

$$3x^{2}-12=x-4+8x$$

Combine x and 8x to get 9x.

$$3x^{2}-12=9x-4$$

Subtract 9x from both sides.

$$3x^{2}-12-9x=-4$$

Add 4 to both sides.

$$3x^{2}-12-9x+4=0$$

Add -12 and 4 to get -8.

$$3x^{2}-8-9x=0$$

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.

$$3x^{2}-9x-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -9 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 3\left(-8\right)}}{2\times 3}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 3\left(-8\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-9\right)±\sqrt{81-12\left(-8\right)}}{2\times 3}$$

Multiply -12 times -8.

$$x=\frac{-\left(-9\right)±\sqrt{81+96}}{2\times 3}$$

Add 81 to 96.

$$x=\frac{-\left(-9\right)±\sqrt{177}}{2\times 3}$$

The opposite of -9 is 9.

$$x=\frac{9±\sqrt{177}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{9±\sqrt{177}}{6}$$

Now solve the equation x=\frac{9±\sqrt{177}}{6} when ± is plus. Add 9 to \sqrt{177}.

$$x=\frac{\sqrt{177}+9}{6}$$

Divide 9+\sqrt{177} by 6.

$$x=\frac{\sqrt{177}}{6}+\frac{3}{2}$$

Now solve the equation x=\frac{9±\sqrt{177}}{6} when ± is minus. Subtract \sqrt{177} from 9.

$$x=\frac{9-\sqrt{177}}{6}$$

Divide 9-\sqrt{177} by 6.

$$x=-\frac{\sqrt{177}}{6}+\frac{3}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{177}}{6}+\frac{3}{2} x=-\frac{\sqrt{177}}{6}+\frac{3}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 3 by x+2.

$$\left(3x+6\right)\left(x-2\right)=x-4+8x$$

Use the distributive property to multiply 3x+6 by x-2 and combine like terms.

$$3x^{2}-12=x-4+8x$$

Combine x and 8x to get 9x.

$$3x^{2}-12=9x-4$$

Subtract 9x from both sides.

$$3x^{2}-12-9x=-4$$

Add 12 to both sides.

$$3x^{2}-9x=-4+12$$

Add -4 and 12 to get 8.

$$3x^{2}-9x=8$$

Divide both sides by 3.

$$\frac{3x^{2}-9x}{3}=\frac{8}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\left(-\frac{9}{3}\right)x=\frac{8}{3}$$

Divide -9 by 3.

$$x^{2}-3x=\frac{8}{3}$$

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=\frac{8}{3}+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=\frac{8}{3}+\frac{9}{4}$$

Add \frac{8}{3} to \frac{9}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-3x+\frac{9}{4}=\frac{59}{12}$$

Factor x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{59}{12}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{59}{12}}$$

Simplify.

$$x-\frac{3}{2}=\frac{\sqrt{177}}{6} x-\frac{3}{2}=-\frac{\sqrt{177}}{6}$$

Add \frac{3}{2} to both sides of the equation.

$$x=\frac{\sqrt{177}}{6}+\frac{3}{2} x=-\frac{\sqrt{177}}{6}+\frac{3}{2}$$