Solve 7+7/w=w-11/w

$$7 + \frac { 7 } { w } = w - \frac { 11 } { w }$$

Solve for w

$$w=-2$$

$$w=9$$

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.

$$w\times 7+7=ww-11$$

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

$$w\times 7+7=w^{2}-11$$

Subtract w^{2} from both sides.

$$w\times 7+7-w^{2}=-11$$

Add 11 to both sides.

$$w\times 7+7-w^{2}+11=0$$

Add 7 and 11 to get 18.

$$w\times 7+18-w^{2}=0$$

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.

$$-w^{2}+7w+18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 7 for b, and 18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-7±\sqrt{7^{2}-4\left(-1\right)\times 18}}{2\left(-1\right)}$$

Square 7.

$$w=\frac{-7±\sqrt{49-4\left(-1\right)\times 18}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$w=\frac{-7±\sqrt{49+4\times 18}}{2\left(-1\right)}$$

Multiply 4 times 18.

$$w=\frac{-7±\sqrt{49+72}}{2\left(-1\right)}$$

Add 49 to 72.

$$w=\frac{-7±\sqrt{121}}{2\left(-1\right)}$$

Take the square root of 121.

$$w=\frac{-7±11}{2\left(-1\right)}$$

Multiply 2 times -1.

$$w=\frac{-7±11}{-2}$$

Now solve the equation w=\frac{-7±11}{-2} when ± is plus. Add -7 to 11.

$$w=\frac{4}{-2}$$

Divide 4 by -2.

$$w=-2$$

Now solve the equation w=\frac{-7±11}{-2} when ± is minus. Subtract 11 from -7.

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

Divide -18 by -2.

$$w=9$$

The equation is now solved.

$$w=-2 w=9$$

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.

$$w\times 7+7=ww-11$$

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

$$w\times 7+7=w^{2}-11$$

Subtract w^{2} from both sides.

$$w\times 7+7-w^{2}=-11$$

Subtract 7 from both sides.

$$w\times 7-w^{2}=-11-7$$

Subtract 7 from -11 to get -18.

$$w\times 7-w^{2}=-18$$

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.

$$-w^{2}+7w=-18$$

Divide both sides by -1.

$$\frac{-w^{2}+7w}{-1}=-\frac{18}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$w^{2}+\frac{7}{-1}w=-\frac{18}{-1}$$

Divide 7 by -1.

$$w^{2}-7w=-\frac{18}{-1}$$

Divide -18 by -1.

$$w^{2}-7w=18$$

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

$$w^{2}-7w+\left(-\frac{7}{2}\right)^{2}=18+\left(-\frac{7}{2}\right)^{2}$$

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$w^{2}-7w+\frac{49}{4}=18+\frac{49}{4}$$

Add 18 to \frac{49}{4}.

$$w^{2}-7w+\frac{49}{4}=\frac{121}{4}$$

Factor w^{2}-7w+\frac{49}{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}.

$$\left(w-\frac{7}{2}\right)^{2}=\frac{121}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(w-\frac{7}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Simplify.

$$w-\frac{7}{2}=\frac{11}{2} w-\frac{7}{2}=-\frac{11}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$w=9 w=-2$$