Solve 16/2x+16-2/2-2x=1

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

Solve for x

$$x=-2$$

$$x=4$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -8,1 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-1\right)\left(x+8\right), the least common multiple of 2x+16,2-2x.

$$\left(x-1\right)\times 16-\left(-\left(8+x\right)\times 2\right)=2\left(x-1\right)\left(x+8\right)$$

Use the distributive property to multiply x-1 by 16.

$$16x-16-\left(-\left(8+x\right)\times 2\right)=2\left(x-1\right)\left(x+8\right)$$

Multiply -1 and 2 to get -2.

$$16x-16-\left(-2\left(8+x\right)\right)=2\left(x-1\right)\left(x+8\right)$$

Use the distributive property to multiply -2 by 8+x.

$$16x-16-\left(-16-2x\right)=2\left(x-1\right)\left(x+8\right)$$

To find the opposite of -16-2x, find the opposite of each term.

$$16x-16+16+2x=2\left(x-1\right)\left(x+8\right)$$

Add -16 and 16 to get 0.

$$16x+2x=2\left(x-1\right)\left(x+8\right)$$

Combine 16x and 2x to get 18x.

$$18x=2\left(x-1\right)\left(x+8\right)$$

Use the distributive property to multiply 2 by x-1.

$$18x=\left(2x-2\right)\left(x+8\right)$$

Use the distributive property to multiply 2x-2 by x+8 and combine like terms.

$$18x=2x^{2}+14x-16$$

Subtract 2x^{2} from both sides.

$$18x-2x^{2}=14x-16$$

Subtract 14x from both sides.

$$18x-2x^{2}-14x=-16$$

Combine 18x and -14x to get 4x.

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

Add 16 to both sides.

$$4x-2x^{2}+16=0$$

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.

$$-2x^{2}+4x+16=0$$

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

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

Square 4.

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

Multiply -4 times -2.

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

Multiply 8 times 16.

$$x=\frac{-4±\sqrt{16+128}}{2\left(-2\right)}$$

Add 16 to 128.

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

Take the square root of 144.

$$x=\frac{-4±12}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-4±12}{-4}$$

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

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

Divide 8 by -4.

$$x=-2$$

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

$$x=-\frac{16}{-4}$$

Divide -16 by -4.

$$x=4$$

The equation is now solved.

$$x=-2 x=4$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -8,1 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-1\right)\left(x+8\right), the least common multiple of 2x+16,2-2x.

$$\left(x-1\right)\times 16-\left(-\left(8+x\right)\times 2\right)=2\left(x-1\right)\left(x+8\right)$$

Use the distributive property to multiply x-1 by 16.

$$16x-16-\left(-\left(8+x\right)\times 2\right)=2\left(x-1\right)\left(x+8\right)$$

Multiply -1 and 2 to get -2.

$$16x-16-\left(-2\left(8+x\right)\right)=2\left(x-1\right)\left(x+8\right)$$

Use the distributive property to multiply -2 by 8+x.

$$16x-16-\left(-16-2x\right)=2\left(x-1\right)\left(x+8\right)$$

To find the opposite of -16-2x, find the opposite of each term.

$$16x-16+16+2x=2\left(x-1\right)\left(x+8\right)$$

Add -16 and 16 to get 0.

$$16x+2x=2\left(x-1\right)\left(x+8\right)$$

Combine 16x and 2x to get 18x.

$$18x=2\left(x-1\right)\left(x+8\right)$$

Use the distributive property to multiply 2 by x-1.

$$18x=\left(2x-2\right)\left(x+8\right)$$

Use the distributive property to multiply 2x-2 by x+8 and combine like terms.

$$18x=2x^{2}+14x-16$$

Subtract 2x^{2} from both sides.

$$18x-2x^{2}=14x-16$$

Subtract 14x from both sides.

$$18x-2x^{2}-14x=-16$$

Combine 18x and -14x to get 4x.

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

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.

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

Divide both sides by -2.

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

Dividing by -2 undoes the multiplication by -2.

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

Divide 4 by -2.

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

Divide -16 by -2.

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

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=8+1$$

Add 8 to 1.

$$x^{2}-2x+1=9$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{9}$$

Simplify.

$$x-1=3 x-1=-3$$

Add 1 to both sides of the equation.

$$x=4 x=-2$$