Solve w+1/w+4=w-5/w+3-1

$$\frac { w + 1 } { w + 4 } = \frac { w - 5 } { w + 3 } - 1$$

Solve for w

$$w=-7$$

$$w=-5$$

Steps Using the Quadratic Formula

Variable w cannot be equal to any of the values -4,-3 since division by zero is not defined. Multiply both sides of the equation by \left(w+3\right)\left(w+4\right), the least common multiple of w+4,w+3.

$$\left(w+3\right)\left(w+1\right)=\left(w+4\right)\left(w-5\right)+\left(w+3\right)\left(w+4\right)\left(-1\right)$$

Use the distributive property to multiply w+3 by w+1 and combine like terms.

$$w^{2}+4w+3=\left(w+4\right)\left(w-5\right)+\left(w+3\right)\left(w+4\right)\left(-1\right)$$

Use the distributive property to multiply w+4 by w-5 and combine like terms.

$$w^{2}+4w+3=w^{2}-w-20+\left(w+3\right)\left(w+4\right)\left(-1\right)$$

Use the distributive property to multiply w+3 by w+4 and combine like terms.

$$w^{2}+4w+3=w^{2}-w-20+\left(w^{2}+7w+12\right)\left(-1\right)$$

Use the distributive property to multiply w^{2}+7w+12 by -1.

$$w^{2}+4w+3=w^{2}-w-20-w^{2}-7w-12$$

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

$$w^{2}+4w+3=-w-20-7w-12$$

Combine -w and -7w to get -8w.

$$w^{2}+4w+3=-8w-20-12$$

Subtract 12 from -20 to get -32.

$$w^{2}+4w+3=-8w-32$$

Add 8w to both sides.

$$w^{2}+4w+3+8w=-32$$

Combine 4w and 8w to get 12w.

$$w^{2}+12w+3=-32$$

Add 32 to both sides.

$$w^{2}+12w+3+32=0$$

Add 3 and 32 to get 35.

$$w^{2}+12w+35=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 12 for b, and 35 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-12±\sqrt{12^{2}-4\times 35}}{2}$$

Square 12.

$$w=\frac{-12±\sqrt{144-4\times 35}}{2}$$

Multiply -4 times 35.

$$w=\frac{-12±\sqrt{144-140}}{2}$$

Add 144 to -140.

$$w=\frac{-12±\sqrt{4}}{2}$$

Take the square root of 4.

$$w=\frac{-12±2}{2}$$

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

$$w=-\frac{10}{2}$$

Divide -10 by 2.

$$w=-5$$

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

$$w=-\frac{14}{2}$$

Divide -14 by 2.

$$w=-7$$

The equation is now solved.

$$w=-5 w=-7$$

Steps for Completing the Square

Variable w cannot be equal to any of the values -4,-3 since division by zero is not defined. Multiply both sides of the equation by \left(w+3\right)\left(w+4\right), the least common multiple of w+4,w+3.

$$\left(w+3\right)\left(w+1\right)=\left(w+4\right)\left(w-5\right)+\left(w+3\right)\left(w+4\right)\left(-1\right)$$

Use the distributive property to multiply w+3 by w+1 and combine like terms.

$$w^{2}+4w+3=\left(w+4\right)\left(w-5\right)+\left(w+3\right)\left(w+4\right)\left(-1\right)$$

Use the distributive property to multiply w+4 by w-5 and combine like terms.

$$w^{2}+4w+3=w^{2}-w-20+\left(w+3\right)\left(w+4\right)\left(-1\right)$$

Use the distributive property to multiply w+3 by w+4 and combine like terms.

$$w^{2}+4w+3=w^{2}-w-20+\left(w^{2}+7w+12\right)\left(-1\right)$$

Use the distributive property to multiply w^{2}+7w+12 by -1.

$$w^{2}+4w+3=w^{2}-w-20-w^{2}-7w-12$$

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

$$w^{2}+4w+3=-w-20-7w-12$$

Combine -w and -7w to get -8w.

$$w^{2}+4w+3=-8w-20-12$$

Subtract 12 from -20 to get -32.

$$w^{2}+4w+3=-8w-32$$

Add 8w to both sides.

$$w^{2}+4w+3+8w=-32$$

Combine 4w and 8w to get 12w.

$$w^{2}+12w+3=-32$$

Subtract 3 from both sides.

$$w^{2}+12w=-32-3$$

Subtract 3 from -32 to get -35.

$$w^{2}+12w=-35$$

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.

$$w^{2}+12w+6^{2}=-35+6^{2}$$

Square 6.

$$w^{2}+12w+36=-35+36$$

Add -35 to 36.

$$w^{2}+12w+36=1$$

Factor w^{2}+12w+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}.

$$\left(w+6\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(w+6\right)^{2}}=\sqrt{1}$$

Simplify.

$$w+6=1 w+6=-1$$

Subtract 6 from both sides of the equation.

$$w=-5 w=-7$$