Solve x^2+4(4x-6/2)=4x

$${ x }^{ 2 } +4( \frac{ 4x-6 }{ 2 } ) = 4x$$

Solve for x

$$x=-6$$

$$x=2$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 2.

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

Use the distributive property to multiply 4 by 4x-6.

$$2x^{2}+16x-24=8x$$

Subtract 8x from both sides.

$$2x^{2}+16x-24-8x=0$$

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

$$2x^{2}+8x-24=0$$

Divide both sides by 2.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-12. To find a and b, set up a system to be solved.

$$a+b=4 ab=1\left(-12\right)=-12$$

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 -12.

$$-1,12 -2,6 -3,4$$

Calculate the sum for each pair.

$$-1+12=11 -2+6=4 -3+4=1$$

The solution is the pair that gives sum 4.

$$a=-2 b=6$$

Rewrite x^{2}+4x-12 as \left(x^{2}-2x\right)+\left(6x-12\right).

$$\left(x^{2}-2x\right)+\left(6x-12\right)$$

Factor out x in the first and 6 in the second group.

$$x\left(x-2\right)+6\left(x-2\right)$$

Factor out common term x-2 by using distributive property.

$$\left(x-2\right)\left(x+6\right)$$

To find equation solutions, solve x-2=0 and x+6=0.

$$x=2 x=-6$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

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

Use the distributive property to multiply 4 by 4x-6.

$$2x^{2}+16x-24=8x$$

Subtract 8x from both sides.

$$2x^{2}+16x-24-8x=0$$

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

$$2x^{2}+8x-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 8 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 2\left(-24\right)}}{2\times 2}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\times 2\left(-24\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-8±\sqrt{64-8\left(-24\right)}}{2\times 2}$$

Multiply -8 times -24.

$$x=\frac{-8±\sqrt{64+192}}{2\times 2}$$

Add 64 to 192.

$$x=\frac{-8±\sqrt{256}}{2\times 2}$$

Take the square root of 256.

$$x=\frac{-8±16}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-8±16}{4}$$

Now solve the equation x=\frac{-8±16}{4} when ± is plus. Add -8 to 16.

$$x=\frac{8}{4}$$

Divide 8 by 4.

$$x=2$$

Now solve the equation x=\frac{-8±16}{4} when ± is minus. Subtract 16 from -8.

$$x=-\frac{24}{4}$$

Divide -24 by 4.

$$x=-6$$

The equation is now solved.

$$x=2 x=-6$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

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

Use the distributive property to multiply 4 by 4x-6.

$$2x^{2}+16x-24=8x$$

Subtract 8x from both sides.

$$2x^{2}+16x-24-8x=0$$

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

$$2x^{2}+8x-24=0$$

Add 24 to both sides. Anything plus zero gives itself.

$$2x^{2}+8x=24$$

Divide both sides by 2.

$$\frac{2x^{2}+8x}{2}=\frac{24}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{8}{2}x=\frac{24}{2}$$

Divide 8 by 2.

$$x^{2}+4x=\frac{24}{2}$$

Divide 24 by 2.

$$x^{2}+4x=12$$

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+4x+2^{2}=12+2^{2}$$

Square 2.

$$x^{2}+4x+4=12+4$$

Add 12 to 4.

$$x^{2}+4x+4=16$$

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}.

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

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{16}$$

Simplify.

$$x+2=4 x+2=-4$$

Subtract 2 from both sides of the equation.

$$x=2 x=-6$$