Solve (x-2)^2+8x=36

$$( x - 2 ) ^ { 2 } + 8 x = 36$$

Solve for x

$$x=-8$$

$$x=4$$

Steps Using Factoring

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

$$x^{2}-4x+4+8x=36$$

Combine -4x and 8x to get 4x.

$$x^{2}+4x+4=36$$

Subtract 36 from both sides.

$$x^{2}+4x+4-36=0$$

Subtract 36 from 4 to get -32.

$$x^{2}+4x-32=0$$

To solve the equation, factor x^{2}+4x-32 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=4 ab=-32$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -32.

$$-1,32 -2,16 -4,8$$

Calculate the sum for each pair.

$$-1+32=31 -2+16=14 -4+8=4$$

The solution is the pair that gives sum 4.

$$a=-4 b=8$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-4\right)\left(x+8\right)$$

To find equation solutions, solve x-4=0 and x+8=0.

$$x=4 x=-8$$

Steps Using Factoring By Grouping

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

$$x^{2}-4x+4+8x=36$$

Combine -4x and 8x to get 4x.

$$x^{2}+4x+4=36$$

Subtract 36 from both sides.

$$x^{2}+4x+4-36=0$$

Subtract 36 from 4 to get -32.

$$x^{2}+4x-32=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-32. To find a and b, set up a system to be solved.

$$a+b=4 ab=1\left(-32\right)=-32$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -32.

$$-1,32 -2,16 -4,8$$

Calculate the sum for each pair.

$$-1+32=31 -2+16=14 -4+8=4$$

The solution is the pair that gives sum 4.

$$a=-4 b=8$$

Rewrite x^{2}+4x-32 as \left(x^{2}-4x\right)+\left(8x-32\right).

$$\left(x^{2}-4x\right)+\left(8x-32\right)$$

Factor out x in the first and 8 in the second group.

$$x\left(x-4\right)+8\left(x-4\right)$$

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

$$\left(x-4\right)\left(x+8\right)$$

To find equation solutions, solve x-4=0 and x+8=0.

$$x=4 x=-8$$

Steps Using the Quadratic Formula

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

$$x^{2}-4x+4+8x=36$$

Combine -4x and 8x to get 4x.

$$x^{2}+4x+4=36$$

Subtract 36 from both sides.

$$x^{2}+4x+4-36=0$$

Subtract 36 from 4 to get -32.

$$x^{2}+4x-32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 4 for b, and -32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-32\right)}}{2}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-32\right)}}{2}$$

Multiply -4 times -32.

$$x=\frac{-4±\sqrt{16+128}}{2}$$

Add 16 to 128.

$$x=\frac{-4±\sqrt{144}}{2}$$

Take the square root of 144.

$$x=\frac{-4±12}{2}$$

Now solve the equation x=\frac{-4±12}{2} when ± is plus. Add -4 to 12.

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

Divide 8 by 2.

$$x=4$$

Now solve the equation x=\frac{-4±12}{2} when ± is minus. Subtract 12 from -4.

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

Divide -16 by 2.

$$x=-8$$

The equation is now solved.

$$x=4 x=-8$$

Steps for Completing the Square

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

$$x^{2}-4x+4+8x=36$$

Combine -4x and 8x to get 4x.

$$x^{2}+4x+4=36$$

Factor x^{2}+4x+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}.

$$\left(x+2\right)^{2}=36$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{36}$$

Simplify.

$$x+2=6 x+2=-6$$

Subtract 2 from both sides of the equation.

$$x=4 x=-8$$