Solve x+4/4-x=4-x+x+4

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

Solve for x

$$x=2\sqrt{2}+6\approx 8.828427125$$

$$x=6-2\sqrt{2}\approx 3.171572875$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 4 since division by zero is not defined. Multiply both sides of the equation by -x+4.

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

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

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

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

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

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

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

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

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

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

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

Combine x^{2} and -x^{2} to get 0.

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

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

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

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

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

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

$$-x^{2}+4x+4=-8x+16+16$$

Add 16 and 16 to get 32.

$$-x^{2}+4x+4=-8x+32$$

Add 8x to both sides.

$$-x^{2}+4x+4+8x=32$$

Combine 4x and 8x to get 12x.

$$-x^{2}+12x+4=32$$

Subtract 32 from both sides.

$$-x^{2}+12x+4-32=0$$

Subtract 32 from 4 to get -28.

$$-x^{2}+12x-28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 12 for b, and -28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-1\right)\left(-28\right)}}{2\left(-1\right)}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\left(-1\right)\left(-28\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-12±\sqrt{144+4\left(-28\right)}}{2\left(-1\right)}$$

Multiply 4 times -28.

$$x=\frac{-12±\sqrt{144-112}}{2\left(-1\right)}$$

Add 144 to -112.

$$x=\frac{-12±\sqrt{32}}{2\left(-1\right)}$$

Take the square root of 32.

$$x=\frac{-12±4\sqrt{2}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-12±4\sqrt{2}}{-2}$$

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

$$x=\frac{4\sqrt{2}-12}{-2}$$

Divide -12+4\sqrt{2} by -2.

$$x=6-2\sqrt{2}$$

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

$$x=\frac{-4\sqrt{2}-12}{-2}$$

Divide -12-4\sqrt{2} by -2.

$$x=2\sqrt{2}+6$$

The equation is now solved.

$$x=6-2\sqrt{2} x=2\sqrt{2}+6$$

Steps for Completing the Square

Variable x cannot be equal to 4 since division by zero is not defined. Multiply both sides of the equation by -x+4.

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

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

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

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

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

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

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

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

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

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

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

Combine x^{2} and -x^{2} to get 0.

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

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

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

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

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

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

$$-x^{2}+4x+4=-8x+16+16$$

Add 16 and 16 to get 32.

$$-x^{2}+4x+4=-8x+32$$

Add 8x to both sides.

$$-x^{2}+4x+4+8x=32$$

Combine 4x and 8x to get 12x.

$$-x^{2}+12x+4=32$$

Subtract 4 from both sides.

$$-x^{2}+12x=32-4$$

Subtract 4 from 32 to get 28.

$$-x^{2}+12x=28$$

Divide both sides by -1.

$$\frac{-x^{2}+12x}{-1}=\frac{28}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{12}{-1}x=\frac{28}{-1}$$

Divide 12 by -1.

$$x^{2}-12x=\frac{28}{-1}$$

Divide 28 by -1.

$$x^{2}-12x=-28$$

Divide -12, the coefficient of the x term, by 2 to get -6. Then add the square of -6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-12x+\left(-6\right)^{2}=-28+\left(-6\right)^{2}$$

Square -6.

$$x^{2}-12x+36=-28+36$$

Add -28 to 36.

$$x^{2}-12x+36=8$$

Factor x^{2}-12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-6=2\sqrt{2} x-6=-2\sqrt{2}$$

Add 6 to both sides of the equation.

$$x=2\sqrt{2}+6 x=6-2\sqrt{2}$$