Solve (2+x)4-(4-x)4=(4-x)(2+x)

$$( 2 + x ) 4 - ( 4 - x ) 4 = ( 4 - x ) ( 2 + x )$$

Solve for x

$$x=-8$$

$$x=2$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2+x by 4.

$$8+4x-\left(4-x\right)\times 4=\left(4-x\right)\left(2+x\right)$$

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

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

To find the opposite of 16-4x, find the opposite of each term.

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

Subtract 16 from 8 to get -8.

$$-8+4x+4x=\left(4-x\right)\left(2+x\right)$$

Combine 4x and 4x to get 8x.

$$-8+8x=\left(4-x\right)\left(2+x\right)$$

Use the distributive property to multiply 4-x by 2+x and combine like terms.

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

Subtract 8 from both sides.

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

Subtract 8 from -8 to get -16.

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

Subtract 2x from both sides.

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

Combine 8x and -2x to get 6x.

$$-16+6x=-x^{2}$$

Add x^{2} to both sides.

$$-16+6x+x^{2}=0$$

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.

$$x^{2}+6x-16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 6 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-16\right)}}{2}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-16\right)}}{2}$$

Multiply -4 times -16.

$$x=\frac{-6±\sqrt{36+64}}{2}$$

Add 36 to 64.

$$x=\frac{-6±\sqrt{100}}{2}$$

Take the square root of 100.

$$x=\frac{-6±10}{2}$$

Now solve the equation x=\frac{-6±10}{2} when ± is plus. Add -6 to 10.

$$x=\frac{4}{2}$$

Divide 4 by 2.

$$x=2$$

Now solve the equation x=\frac{-6±10}{2} when ± is minus. Subtract 10 from -6.

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

Divide -16 by 2.

$$x=-8$$

The equation is now solved.

$$x=2 x=-8$$

Steps for Completing the Square

Use the distributive property to multiply 2+x by 4.

$$8+4x-\left(4-x\right)\times 4=\left(4-x\right)\left(2+x\right)$$

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

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

To find the opposite of 16-4x, find the opposite of each term.

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

Subtract 16 from 8 to get -8.

$$-8+4x+4x=\left(4-x\right)\left(2+x\right)$$

Combine 4x and 4x to get 8x.

$$-8+8x=\left(4-x\right)\left(2+x\right)$$

Use the distributive property to multiply 4-x by 2+x and combine like terms.

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

Subtract 2x from both sides.

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

Combine 8x and -2x to get 6x.

$$-8+6x=8-x^{2}$$

Add x^{2} to both sides.

$$-8+6x+x^{2}=8$$

Add 8 to both sides.

$$6x+x^{2}=8+8$$

Add 8 and 8 to get 16.

$$6x+x^{2}=16$$

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.

$$x^{2}+6x=16$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+6x+3^{2}=16+3^{2}$$

Square 3.

$$x^{2}+6x+9=16+9$$

Add 16 to 9.

$$x^{2}+6x+9=25$$

Factor x^{2}+6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+3\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{25}$$

Simplify.

$$x+3=5 x+3=-5$$

Subtract 3 from both sides of the equation.

$$x=2 x=-8$$