Solve x^2-frac{(2sqrt{6}x+6sqrt{6})^2}{8}=1

$$x ^ { 2 } - \frac { ( 2 \sqrt { 6 } x + 6 \sqrt { 6 } ) ^ { 2 } } { 8 } = 1$$

Solve for x

$$x=-7$$

$$x=-2$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 8.

$$8x^{2}-\left(2\sqrt{6}x+6\sqrt{6}\right)^{2}=8$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2\sqrt{6}x+6\sqrt{6}\right)^{2}.

$$8x^{2}-\left(4\left(\sqrt{6}\right)^{2}x^{2}+24\sqrt{6}x\sqrt{6}+36\left(\sqrt{6}\right)^{2}\right)=8$$

Multiply \sqrt{6} and \sqrt{6} to get 6.

$$8x^{2}-\left(4\left(\sqrt{6}\right)^{2}x^{2}+24\times 6x+36\left(\sqrt{6}\right)^{2}\right)=8$$

The square of \sqrt{6} is 6.

$$8x^{2}-\left(4\times 6x^{2}+24\times 6x+36\left(\sqrt{6}\right)^{2}\right)=8$$

Multiply 4 and 6 to get 24.

$$8x^{2}-\left(24x^{2}+24\times 6x+36\left(\sqrt{6}\right)^{2}\right)=8$$

Multiply 24 and 6 to get 144.

$$8x^{2}-\left(24x^{2}+144x+36\left(\sqrt{6}\right)^{2}\right)=8$$

The square of \sqrt{6} is 6.

$$8x^{2}-\left(24x^{2}+144x+36\times 6\right)=8$$

Multiply 36 and 6 to get 216.

$$8x^{2}-\left(24x^{2}+144x+216\right)=8$$

To find the opposite of 24x^{2}+144x+216, find the opposite of each term.

$$8x^{2}-24x^{2}-144x-216=8$$

Combine 8x^{2} and -24x^{2} to get -16x^{2}.

$$-16x^{2}-144x-216=8$$

Subtract 8 from both sides.

$$-16x^{2}-144x-216-8=0$$

Subtract 8 from -216 to get -224.

$$-16x^{2}-144x-224=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, -144 for b, and -224 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-144\right)±\sqrt{\left(-144\right)^{2}-4\left(-16\right)\left(-224\right)}}{2\left(-16\right)}$$

Square -144.

$$x=\frac{-\left(-144\right)±\sqrt{20736-4\left(-16\right)\left(-224\right)}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-\left(-144\right)±\sqrt{20736+64\left(-224\right)}}{2\left(-16\right)}$$

Multiply 64 times -224.

$$x=\frac{-\left(-144\right)±\sqrt{20736-14336}}{2\left(-16\right)}$$

Add 20736 to -14336.

$$x=\frac{-\left(-144\right)±\sqrt{6400}}{2\left(-16\right)}$$

Take the square root of 6400.

$$x=\frac{-\left(-144\right)±80}{2\left(-16\right)}$$

The opposite of -144 is 144.

$$x=\frac{144±80}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{144±80}{-32}$$

Now solve the equation x=\frac{144±80}{-32} when ± is plus. Add 144 to 80.

$$x=\frac{224}{-32}$$

Divide 224 by -32.

$$x=-7$$

Now solve the equation x=\frac{144±80}{-32} when ± is minus. Subtract 80 from 144.

$$x=\frac{64}{-32}$$

Divide 64 by -32.

$$x=-2$$

The equation is now solved.

$$x=-7 x=-2$$

Steps for Completing the Square

Multiply both sides of the equation by 8.

$$8x^{2}-\left(2\sqrt{6}x+6\sqrt{6}\right)^{2}=8$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2\sqrt{6}x+6\sqrt{6}\right)^{2}.

$$8x^{2}-\left(4\left(\sqrt{6}\right)^{2}x^{2}+24\sqrt{6}x\sqrt{6}+36\left(\sqrt{6}\right)^{2}\right)=8$$

Multiply \sqrt{6} and \sqrt{6} to get 6.

$$8x^{2}-\left(4\left(\sqrt{6}\right)^{2}x^{2}+24\times 6x+36\left(\sqrt{6}\right)^{2}\right)=8$$

The square of \sqrt{6} is 6.

$$8x^{2}-\left(4\times 6x^{2}+24\times 6x+36\left(\sqrt{6}\right)^{2}\right)=8$$

Multiply 4 and 6 to get 24.

$$8x^{2}-\left(24x^{2}+24\times 6x+36\left(\sqrt{6}\right)^{2}\right)=8$$

Multiply 24 and 6 to get 144.

$$8x^{2}-\left(24x^{2}+144x+36\left(\sqrt{6}\right)^{2}\right)=8$$

The square of \sqrt{6} is 6.

$$8x^{2}-\left(24x^{2}+144x+36\times 6\right)=8$$

Multiply 36 and 6 to get 216.

$$8x^{2}-\left(24x^{2}+144x+216\right)=8$$

To find the opposite of 24x^{2}+144x+216, find the opposite of each term.

$$8x^{2}-24x^{2}-144x-216=8$$

Combine 8x^{2} and -24x^{2} to get -16x^{2}.

$$-16x^{2}-144x-216=8$$

Add 216 to both sides.

$$-16x^{2}-144x=8+216$$

Add 8 and 216 to get 224.

$$-16x^{2}-144x=224$$

Divide both sides by -16.

$$\frac{-16x^{2}-144x}{-16}=\frac{224}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$x^{2}+\left(-\frac{144}{-16}\right)x=\frac{224}{-16}$$

Divide -144 by -16.

$$x^{2}+9x=\frac{224}{-16}$$

Divide 224 by -16.

$$x^{2}+9x=-14$$

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.

$$x^{2}+9x+\left(\frac{9}{2}\right)^{2}=-14+\left(\frac{9}{2}\right)^{2}$$

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

$$x^{2}+9x+\frac{81}{4}=-14+\frac{81}{4}$$

Add -14 to \frac{81}{4}.

$$x^{2}+9x+\frac{81}{4}=\frac{25}{4}$$

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}.

$$\left(x+\frac{9}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{9}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$x+\frac{9}{2}=\frac{5}{2} x+\frac{9}{2}=-\frac{5}{2}$$

Subtract \frac{9}{2} from both sides of the equation.

$$x=-2 x=-7$$