Solve -5-6/w-7=-7/(w+1)(w-7)

$$- 5 - \frac { 6 } { w - 7 } = - \frac { 7 } { ( w + 1 ) ( w - 7 ) }$$

Solve for w

$$w = -\frac{6}{5} = -1\frac{1}{5} = -1.2$$

$$w=6$$

Steps Using the Quadratic Formula

Variable w cannot be equal to any of the values -1,7 since division by zero is not defined. Multiply both sides of the equation by \left(w-7\right)\left(w+1\right), the least common multiple of w-7,\left(w+1\right)\left(w-7\right).

$$\left(w-7\right)\left(w+1\right)\left(-5\right)-\left(w+1\right)\times 6=-7$$

Use the distributive property to multiply w-7 by w+1 and combine like terms.

$$\left(w^{2}-6w-7\right)\left(-5\right)-\left(w+1\right)\times 6=-7$$

Use the distributive property to multiply w^{2}-6w-7 by -5.

$$-5w^{2}+30w+35-\left(w+1\right)\times 6=-7$$

Use the distributive property to multiply w+1 by 6.

$$-5w^{2}+30w+35-\left(6w+6\right)=-7$$

To find the opposite of 6w+6, find the opposite of each term.

$$-5w^{2}+30w+35-6w-6=-7$$

Combine 30w and -6w to get 24w.

$$-5w^{2}+24w+35-6=-7$$

Subtract 6 from 35 to get 29.

$$-5w^{2}+24w+29=-7$$

Add 7 to both sides.

$$-5w^{2}+24w+29+7=0$$

Add 29 and 7 to get 36.

$$-5w^{2}+24w+36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 24 for b, and 36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-24±\sqrt{24^{2}-4\left(-5\right)\times 36}}{2\left(-5\right)}$$

Square 24.

$$w=\frac{-24±\sqrt{576-4\left(-5\right)\times 36}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$w=\frac{-24±\sqrt{576+20\times 36}}{2\left(-5\right)}$$

Multiply 20 times 36.

$$w=\frac{-24±\sqrt{576+720}}{2\left(-5\right)}$$

Add 576 to 720.

$$w=\frac{-24±\sqrt{1296}}{2\left(-5\right)}$$

Take the square root of 1296.

$$w=\frac{-24±36}{2\left(-5\right)}$$

Multiply 2 times -5.

$$w=\frac{-24±36}{-10}$$

Now solve the equation w=\frac{-24±36}{-10} when ± is plus. Add -24 to 36.

$$w=\frac{12}{-10}$$

Reduce the fraction \frac{12}{-10} to lowest terms by extracting and canceling out 2.

$$w=-\frac{6}{5}$$

Now solve the equation w=\frac{-24±36}{-10} when ± is minus. Subtract 36 from -24.

$$w=-\frac{60}{-10}$$

Divide -60 by -10.

$$w=6$$

The equation is now solved.

$$w=-\frac{6}{5} w=6$$

Steps for Completing the Square

Variable w cannot be equal to any of the values -1,7 since division by zero is not defined. Multiply both sides of the equation by \left(w-7\right)\left(w+1\right), the least common multiple of w-7,\left(w+1\right)\left(w-7\right).

$$\left(w-7\right)\left(w+1\right)\left(-5\right)-\left(w+1\right)\times 6=-7$$

Use the distributive property to multiply w-7 by w+1 and combine like terms.

$$\left(w^{2}-6w-7\right)\left(-5\right)-\left(w+1\right)\times 6=-7$$

Use the distributive property to multiply w^{2}-6w-7 by -5.

$$-5w^{2}+30w+35-\left(w+1\right)\times 6=-7$$

Use the distributive property to multiply w+1 by 6.

$$-5w^{2}+30w+35-\left(6w+6\right)=-7$$

To find the opposite of 6w+6, find the opposite of each term.

$$-5w^{2}+30w+35-6w-6=-7$$

Combine 30w and -6w to get 24w.

$$-5w^{2}+24w+35-6=-7$$

Subtract 6 from 35 to get 29.

$$-5w^{2}+24w+29=-7$$

Subtract 29 from both sides.

$$-5w^{2}+24w=-7-29$$

Subtract 29 from -7 to get -36.

$$-5w^{2}+24w=-36$$

Divide both sides by -5.

$$\frac{-5w^{2}+24w}{-5}=-\frac{36}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$w^{2}+\frac{24}{-5}w=-\frac{36}{-5}$$

Divide 24 by -5.

$$w^{2}-\frac{24}{5}w=-\frac{36}{-5}$$

Divide -36 by -5.

$$w^{2}-\frac{24}{5}w=\frac{36}{5}$$

Divide -\frac{24}{5}, the coefficient of the x term, by 2 to get -\frac{12}{5}. Then add the square of -\frac{12}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$w^{2}-\frac{24}{5}w+\left(-\frac{12}{5}\right)^{2}=\frac{36}{5}+\left(-\frac{12}{5}\right)^{2}$$

Square -\frac{12}{5} by squaring both the numerator and the denominator of the fraction.

$$w^{2}-\frac{24}{5}w+\frac{144}{25}=\frac{36}{5}+\frac{144}{25}$$

Add \frac{36}{5} to \frac{144}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$w^{2}-\frac{24}{5}w+\frac{144}{25}=\frac{324}{25}$$

Factor w^{2}-\frac{24}{5}w+\frac{144}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(w-\frac{12}{5}\right)^{2}=\frac{324}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(w-\frac{12}{5}\right)^{2}}=\sqrt{\frac{324}{25}}$$

Simplify.

$$w-\frac{12}{5}=\frac{18}{5} w-\frac{12}{5}=-\frac{18}{5}$$

Add \frac{12}{5} to both sides of the equation.

$$w=6 w=-\frac{6}{5}$$