Solve x^2+2=8x(x-2)

$$x ^ { 2 } + 2 = 8 x ( x - 2 )$$

Solve for x

$$x = \frac{\sqrt{78} + 8}{7} \approx 2.404537267$$

$$x=\frac{8-\sqrt{78}}{7}\approx -0.118822981$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 8x by x-2.

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

Subtract 8x^{2} from both sides.

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

Combine x^{2} and -8x^{2} to get -7x^{2}.

$$-7x^{2}+2=-16x$$

Add 16x to both sides.

$$-7x^{2}+2+16x=0$$

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.

$$-7x^{2}+16x+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, 16 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-7\right)\times 2}}{2\left(-7\right)}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-7\right)\times 2}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-16±\sqrt{256+28\times 2}}{2\left(-7\right)}$$

Multiply 28 times 2.

$$x=\frac{-16±\sqrt{256+56}}{2\left(-7\right)}$$

Add 256 to 56.

$$x=\frac{-16±\sqrt{312}}{2\left(-7\right)}$$

Take the square root of 312.

$$x=\frac{-16±2\sqrt{78}}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{-16±2\sqrt{78}}{-14}$$

Now solve the equation x=\frac{-16±2\sqrt{78}}{-14} when ± is plus. Add -16 to 2\sqrt{78}.

$$x=\frac{2\sqrt{78}-16}{-14}$$

Divide -16+2\sqrt{78} by -14.

$$x=\frac{8-\sqrt{78}}{7}$$

Now solve the equation x=\frac{-16±2\sqrt{78}}{-14} when ± is minus. Subtract 2\sqrt{78} from -16.

$$x=\frac{-2\sqrt{78}-16}{-14}$$

Divide -16-2\sqrt{78} by -14.

$$x=\frac{\sqrt{78}+8}{7}$$

The equation is now solved.

$$x=\frac{8-\sqrt{78}}{7} x=\frac{\sqrt{78}+8}{7}$$

Steps for Completing the Square

Use the distributive property to multiply 8x by x-2.

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

Subtract 8x^{2} from both sides.

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

Combine x^{2} and -8x^{2} to get -7x^{2}.

$$-7x^{2}+2=-16x$$

Add 16x to both sides.

$$-7x^{2}+2+16x=0$$

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

$$-7x^{2}+16x=-2$$

Divide both sides by -7.

$$\frac{-7x^{2}+16x}{-7}=-\frac{2}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$x^{2}+\frac{16}{-7}x=-\frac{2}{-7}$$

Divide 16 by -7.

$$x^{2}-\frac{16}{7}x=-\frac{2}{-7}$$

Divide -2 by -7.

$$x^{2}-\frac{16}{7}x=\frac{2}{7}$$

Divide -\frac{16}{7}, the coefficient of the x term, by 2 to get -\frac{8}{7}. Then add the square of -\frac{8}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{16}{7}x+\left(-\frac{8}{7}\right)^{2}=\frac{2}{7}+\left(-\frac{8}{7}\right)^{2}$$

Square -\frac{8}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{16}{7}x+\frac{64}{49}=\frac{2}{7}+\frac{64}{49}$$

Add \frac{2}{7} to \frac{64}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{16}{7}x+\frac{64}{49}=\frac{78}{49}$$

Factor x^{2}-\frac{16}{7}x+\frac{64}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{8}{7}\right)^{2}=\frac{78}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{8}{7}\right)^{2}}=\sqrt{\frac{78}{49}}$$

Simplify.

$$x-\frac{8}{7}=\frac{\sqrt{78}}{7} x-\frac{8}{7}=-\frac{\sqrt{78}}{7}$$

Add \frac{8}{7} to both sides of the equation.

$$x=\frac{\sqrt{78}+8}{7} x=\frac{8-\sqrt{78}}{7}$$