Solve w/5=w-3/w+1+12/5w+5

$$\frac { w } { 5 } = \frac { w - 3 } { w + 1 } + \frac { 12 } { 5 w + 5 }$$

Solve for w

$$w=1$$

$$w=3$$

Steps Using the Quadratic Formula

Variable w cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by 5\left(w+1\right), the least common multiple of 5,w+1,5w+5.

$$\left(w+1\right)w=5\left(w-3\right)+12$$

Use the distributive property to multiply w+1 by w.

$$w^{2}+w=5\left(w-3\right)+12$$

Use the distributive property to multiply 5 by w-3.

$$w^{2}+w=5w-15+12$$

Add -15 and 12 to get -3.

$$w^{2}+w=5w-3$$

Subtract 5w from both sides.

$$w^{2}+w-5w=-3$$

Combine w and -5w to get -4w.

$$w^{2}-4w=-3$$

Add 3 to both sides.

$$w^{2}-4w+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 3}}{2}$$

Square -4.

$$w=\frac{-\left(-4\right)±\sqrt{16-4\times 3}}{2}$$

Multiply -4 times 3.

$$w=\frac{-\left(-4\right)±\sqrt{16-12}}{2}$$

Add 16 to -12.

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

Take the square root of 4.

$$w=\frac{-\left(-4\right)±2}{2}$$

The opposite of -4 is 4.

$$w=\frac{4±2}{2}$$

Now solve the equation w=\frac{4±2}{2} when ± is plus. Add 4 to 2.

$$w=\frac{6}{2}$$

Divide 6 by 2.

$$w=3$$

Now solve the equation w=\frac{4±2}{2} when ± is minus. Subtract 2 from 4.

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

Divide 2 by 2.

$$w=1$$

The equation is now solved.

$$w=3 w=1$$

Steps for Completing the Square

Variable w cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by 5\left(w+1\right), the least common multiple of 5,w+1,5w+5.

$$\left(w+1\right)w=5\left(w-3\right)+12$$

Use the distributive property to multiply w+1 by w.

$$w^{2}+w=5\left(w-3\right)+12$$

Use the distributive property to multiply 5 by w-3.

$$w^{2}+w=5w-15+12$$

Add -15 and 12 to get -3.

$$w^{2}+w=5w-3$$

Subtract 5w from both sides.

$$w^{2}+w-5w=-3$$

Combine w and -5w to get -4w.

$$w^{2}-4w=-3$$

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.

$$w^{2}-4w+\left(-2\right)^{2}=-3+\left(-2\right)^{2}$$

Square -2.

$$w^{2}-4w+4=-3+4$$

Add -3 to 4.

$$w^{2}-4w+4=1$$

Factor w^{2}-4w+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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$w-2=1 w-2=-1$$

Add 2 to both sides of the equation.

$$w=3 w=1$$