Solve -12+8/w=w+35/w

$$- 12 + \frac { 8 } { w } = w + \frac { 35 } { w }$$

Solve for w

$$w=-9$$

$$w=-3$$

Steps Using the Quadratic Formula

Variable w cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by w.

$$w\left(-12\right)+8=ww+35$$

Multiply w and w to get w^{2}.

$$w\left(-12\right)+8=w^{2}+35$$

Subtract w^{2} from both sides.

$$w\left(-12\right)+8-w^{2}=35$$

Subtract 35 from both sides.

$$w\left(-12\right)+8-w^{2}-35=0$$

Subtract 35 from 8 to get -27.

$$w\left(-12\right)-27-w^{2}=0$$

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.

$$-w^{2}-12w-27=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -12 for b, and -27 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-1\right)\left(-27\right)}}{2\left(-1\right)}$$

Square -12.

$$w=\frac{-\left(-12\right)±\sqrt{144-4\left(-1\right)\left(-27\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$w=\frac{-\left(-12\right)±\sqrt{144+4\left(-27\right)}}{2\left(-1\right)}$$

Multiply 4 times -27.

$$w=\frac{-\left(-12\right)±\sqrt{144-108}}{2\left(-1\right)}$$

Add 144 to -108.

$$w=\frac{-\left(-12\right)±\sqrt{36}}{2\left(-1\right)}$$

Take the square root of 36.

$$w=\frac{-\left(-12\right)±6}{2\left(-1\right)}$$

The opposite of -12 is 12.

$$w=\frac{12±6}{2\left(-1\right)}$$

Multiply 2 times -1.

$$w=\frac{12±6}{-2}$$

Now solve the equation w=\frac{12±6}{-2} when ± is plus. Add 12 to 6.

$$w=\frac{18}{-2}$$

Divide 18 by -2.

$$w=-9$$

Now solve the equation w=\frac{12±6}{-2} when ± is minus. Subtract 6 from 12.

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

Divide 6 by -2.

$$w=-3$$

The equation is now solved.

$$w=-9 w=-3$$

Steps for Completing the Square

Variable w cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by w.

$$w\left(-12\right)+8=ww+35$$

Multiply w and w to get w^{2}.

$$w\left(-12\right)+8=w^{2}+35$$

Subtract w^{2} from both sides.

$$w\left(-12\right)+8-w^{2}=35$$

Subtract 8 from both sides.

$$w\left(-12\right)-w^{2}=35-8$$

Subtract 8 from 35 to get 27.

$$w\left(-12\right)-w^{2}=27$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-w^{2}-12w=27$$

Divide both sides by -1.

$$\frac{-w^{2}-12w}{-1}=\frac{27}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$w^{2}+\left(-\frac{12}{-1}\right)w=\frac{27}{-1}$$

Divide -12 by -1.

$$w^{2}+12w=\frac{27}{-1}$$

Divide 27 by -1.

$$w^{2}+12w=-27$$

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.

$$w^{2}+12w+6^{2}=-27+6^{2}$$

Square 6.

$$w^{2}+12w+36=-27+36$$

Add -27 to 36.

$$w^{2}+12w+36=9$$

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}.

$$\left(w+6\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(w+6\right)^{2}}=\sqrt{9}$$

Simplify.

$$w+6=3 w+6=-3$$

Subtract 6 from both sides of the equation.

$$w=-3 w=-9$$