Solve 2-3/w+7=-7/(w-2)(w+7)

$$2 - \frac { 3 } { w + 7 } = - \frac { 7 } { ( w - 2 ) ( w + 7 ) }$$

Solve for w

$$w=-5$$

$$w = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Steps Using the Quadratic Formula

Variable w cannot be equal to any of the values -7,2 since division by zero is not defined. Multiply both sides of the equation by \left(w-2\right)\left(w+7\right), the least common multiple of w+7,\left(w-2\right)\left(w+7\right).

$$\left(w-2\right)\left(w+7\right)\times 2-\left(w-2\right)\times 3=-7$$

Use the distributive property to multiply w-2 by w+7 and combine like terms.

$$\left(w^{2}+5w-14\right)\times 2-\left(w-2\right)\times 3=-7$$

Use the distributive property to multiply w^{2}+5w-14 by 2.

$$2w^{2}+10w-28-\left(w-2\right)\times 3=-7$$

Use the distributive property to multiply w-2 by 3.

$$2w^{2}+10w-28-\left(3w-6\right)=-7$$

To find the opposite of 3w-6, find the opposite of each term.

$$2w^{2}+10w-28-3w+6=-7$$

Combine 10w and -3w to get 7w.

$$2w^{2}+7w-28+6=-7$$

Add -28 and 6 to get -22.

$$2w^{2}+7w-22=-7$$

Add 7 to both sides.

$$2w^{2}+7w-22+7=0$$

Add -22 and 7 to get -15.

$$2w^{2}+7w-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 7 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-7±\sqrt{7^{2}-4\times 2\left(-15\right)}}{2\times 2}$$

Square 7.

$$w=\frac{-7±\sqrt{49-4\times 2\left(-15\right)}}{2\times 2}$$

Multiply -4 times 2.

$$w=\frac{-7±\sqrt{49-8\left(-15\right)}}{2\times 2}$$

Multiply -8 times -15.

$$w=\frac{-7±\sqrt{49+120}}{2\times 2}$$

Add 49 to 120.

$$w=\frac{-7±\sqrt{169}}{2\times 2}$$

Take the square root of 169.

$$w=\frac{-7±13}{2\times 2}$$

Multiply 2 times 2.

$$w=\frac{-7±13}{4}$$

Now solve the equation w=\frac{-7±13}{4} when ± is plus. Add -7 to 13.

$$w=\frac{6}{4}$$

Reduce the fraction \frac{6}{4} to lowest terms by extracting and canceling out 2.

$$w=\frac{3}{2}$$

Now solve the equation w=\frac{-7±13}{4} when ± is minus. Subtract 13 from -7.

$$w=-\frac{20}{4}$$

Divide -20 by 4.

$$w=-5$$

The equation is now solved.

$$w=\frac{3}{2} w=-5$$

Steps for Completing the Square

Variable w cannot be equal to any of the values -7,2 since division by zero is not defined. Multiply both sides of the equation by \left(w-2\right)\left(w+7\right), the least common multiple of w+7,\left(w-2\right)\left(w+7\right).

$$\left(w-2\right)\left(w+7\right)\times 2-\left(w-2\right)\times 3=-7$$

Use the distributive property to multiply w-2 by w+7 and combine like terms.

$$\left(w^{2}+5w-14\right)\times 2-\left(w-2\right)\times 3=-7$$

Use the distributive property to multiply w^{2}+5w-14 by 2.

$$2w^{2}+10w-28-\left(w-2\right)\times 3=-7$$

Use the distributive property to multiply w-2 by 3.

$$2w^{2}+10w-28-\left(3w-6\right)=-7$$

To find the opposite of 3w-6, find the opposite of each term.

$$2w^{2}+10w-28-3w+6=-7$$

Combine 10w and -3w to get 7w.

$$2w^{2}+7w-28+6=-7$$

Add -28 and 6 to get -22.

$$2w^{2}+7w-22=-7$$

Add 22 to both sides.

$$2w^{2}+7w=-7+22$$

Add -7 and 22 to get 15.

$$2w^{2}+7w=15$$

Divide both sides by 2.

$$\frac{2w^{2}+7w}{2}=\frac{15}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$w^{2}+\frac{7}{2}w=\frac{15}{2}$$

Divide \frac{7}{2}, the coefficient of the x term, by 2 to get \frac{7}{4}. Then add the square of \frac{7}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$w^{2}+\frac{7}{2}w+\left(\frac{7}{4}\right)^{2}=\frac{15}{2}+\left(\frac{7}{4}\right)^{2}$$

Square \frac{7}{4} by squaring both the numerator and the denominator of the fraction.

$$w^{2}+\frac{7}{2}w+\frac{49}{16}=\frac{15}{2}+\frac{49}{16}$$

Add \frac{15}{2} to \frac{49}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$w^{2}+\frac{7}{2}w+\frac{49}{16}=\frac{169}{16}$$

Factor w^{2}+\frac{7}{2}w+\frac{49}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(w+\frac{7}{4}\right)^{2}=\frac{169}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(w+\frac{7}{4}\right)^{2}}=\sqrt{\frac{169}{16}}$$

Simplify.

$$w+\frac{7}{4}=\frac{13}{4} w+\frac{7}{4}=-\frac{13}{4}$$

Subtract \frac{7}{4} from both sides of the equation.

$$w=\frac{3}{2} w=-5$$