Solve w^2/w-4-8/w-4=2w/w-4

$$\frac { w ^ { 2 } } { w - 4 } - \frac { 8 } { w - 4 } = \frac { 2 w } { w - 4 }$$

Solve for w

$$w=-2$$

Steps Using Factoring

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

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

Subtract 2w from both sides.

$$w^{2}-8-2w=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$w^{2}-2w-8=0$$

To solve the equation, factor w^{2}-2w-8 using formula w^{2}+\left(a+b\right)w+ab=\left(w+a\right)\left(w+b\right). To find a and b, set up a system to be solved.

$$a+b=-2 ab=-8$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -8.

$$1,-8 2,-4$$

Calculate the sum for each pair.

$$1-8=-7 2-4=-2$$

The solution is the pair that gives sum -2.

$$a=-4 b=2$$

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

$$\left(w-4\right)\left(w+2\right)$$

To find equation solutions, solve w-4=0 and w+2=0.

$$w=4 w=-2$$

Variable w cannot be equal to 4.

$$w=-2$$

Steps Using Factoring By Grouping

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

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

Subtract 2w from both sides.

$$w^{2}-8-2w=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$w^{2}-2w-8=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as w^{2}+aw+bw-8. To find a and b, set up a system to be solved.

$$a+b=-2 ab=1\left(-8\right)=-8$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -8.

$$1,-8 2,-4$$

Calculate the sum for each pair.

$$1-8=-7 2-4=-2$$

The solution is the pair that gives sum -2.

$$a=-4 b=2$$

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

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

Factor out w in the first and 2 in the second group.

$$w\left(w-4\right)+2\left(w-4\right)$$

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

$$\left(w-4\right)\left(w+2\right)$$

To find equation solutions, solve w-4=0 and w+2=0.

$$w=4 w=-2$$

Variable w cannot be equal to 4.

$$w=-2$$

Steps Using the Quadratic Formula

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

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

Subtract 2w from both sides.

$$w^{2}-8-2w=0$$

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.

$$w^{2}-2w-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -2.

$$w=\frac{-\left(-2\right)±\sqrt{4-4\left(-8\right)}}{2}$$

Multiply -4 times -8.

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

Add 4 to 32.

$$w=\frac{-\left(-2\right)±\sqrt{36}}{2}$$

Take the square root of 36.

$$w=\frac{-\left(-2\right)±6}{2}$$

The opposite of -2 is 2.

$$w=\frac{2±6}{2}$$

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

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

Divide 8 by 2.

$$w=4$$

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

$$w=-\frac{4}{2}$$

Divide -4 by 2.

$$w=-2$$

The equation is now solved.

$$w=4 w=-2$$

Variable w cannot be equal to 4.

$$w=-2$$

Steps for Completing the Square

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

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

Subtract 2w from both sides.

$$w^{2}-8-2w=0$$

Add 8 to both sides. Anything plus zero gives itself.

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

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.

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

Add 8 to 1.

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

Factor w^{2}-2w+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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$w-1=3 w-1=-3$$

Add 1 to both sides of the equation.

$$w=4 w=-2$$

Variable w cannot be equal to 4.

$$w=-2$$