Solve 16x-16=x^2+8x

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

Solve for x

$$x=4$$

Steps Using Factoring By Grouping

Subtract x^{2} from both sides.

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

Subtract 8x from both sides.

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

Combine 16x and -8x to get 8x.

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

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}+8x-16=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-16. To find a and b, set up a system to be solved.

$$a+b=8 ab=-\left(-16\right)=16$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 16.

$$1,16 2,8 4,4$$

Calculate the sum for each pair.

$$1+16=17 2+8=10 4+4=8$$

The solution is the pair that gives sum 8.

$$a=4 b=4$$

Rewrite -x^{2}+8x-16 as \left(-x^{2}+4x\right)+\left(4x-16\right).

$$\left(-x^{2}+4x\right)+\left(4x-16\right)$$

Factor out -x in the first and 4 in the second group.

$$-x\left(x-4\right)+4\left(x-4\right)$$

Factor out common term x-4 by using distributive property.

$$\left(x-4\right)\left(-x+4\right)$$

To find equation solutions, solve x-4=0 and -x+4=0.

$$x=4 x=4$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

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

Subtract 8x from both sides.

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

Combine 16x and -8x to get 8x.

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

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.

$$-x^{2}+8x-16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 8 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-1\right)\left(-16\right)}}{2\left(-1\right)}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-1\right)\left(-16\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-8±\sqrt{64+4\left(-16\right)}}{2\left(-1\right)}$$

Multiply 4 times -16.

$$x=\frac{-8±\sqrt{64-64}}{2\left(-1\right)}$$

Add 64 to -64.

$$x=\frac{-8±\sqrt{0}}{2\left(-1\right)}$$

Take the square root of 0.

$$x=-\frac{8}{2\left(-1\right)}$$

Multiply 2 times -1.

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

Divide -8 by -2.

$$x=4$$

Steps for Completing the Square

Subtract x^{2} from both sides.

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

Subtract 8x from both sides.

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

Combine 16x and -8x to get 8x.

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

Add 16 to both sides. Anything plus zero gives itself.

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

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.

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

Divide both sides by -1.

$$\frac{-x^{2}+8x}{-1}=\frac{16}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{8}{-1}x=\frac{16}{-1}$$

Divide 8 by -1.

$$x^{2}-8x=\frac{16}{-1}$$

Divide 16 by -1.

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

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.

$$x^{2}-8x+\left(-4\right)^{2}=-16+\left(-4\right)^{2}$$

Square -4.

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

Add -16 to 16.

$$x^{2}-8x+16=0$$

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}.

$$\left(x-4\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-4=0 x-4=0$$

Add 4 to both sides of the equation.

$$x=4 x=4$$

The equation is now solved. Solutions are the same.

$$x=4$$