Solve p-2/p+2=13

$$p - \frac { 2 } { p + 2 } = 13$$

Solve for p

$$p = \frac{\sqrt{233} + 11}{2} \approx 13.132168761$$

$$p=\frac{11-\sqrt{233}}{2}\approx -2.132168761$$

Steps Using the Quadratic Formula

Variable p cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by p+2.

$$\left(p+2\right)p-2=13\left(p+2\right)$$

Use the distributive property to multiply p+2 by p.

$$p^{2}+2p-2=13\left(p+2\right)$$

Use the distributive property to multiply 13 by p+2.

$$p^{2}+2p-2=13p+26$$

Subtract 13p from both sides.

$$p^{2}+2p-2-13p=26$$

Combine 2p and -13p to get -11p.

$$p^{2}-11p-2=26$$

Subtract 26 from both sides.

$$p^{2}-11p-2-26=0$$

Subtract 26 from -2 to get -28.

$$p^{2}-11p-28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -11 for b, and -28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\left(-28\right)}}{2}$$

Square -11.

$$p=\frac{-\left(-11\right)±\sqrt{121-4\left(-28\right)}}{2}$$

Multiply -4 times -28.

$$p=\frac{-\left(-11\right)±\sqrt{121+112}}{2}$$

Add 121 to 112.

$$p=\frac{-\left(-11\right)±\sqrt{233}}{2}$$

The opposite of -11 is 11.

$$p=\frac{11±\sqrt{233}}{2}$$

Now solve the equation p=\frac{11±\sqrt{233}}{2} when ± is plus. Add 11 to \sqrt{233}.

$$p=\frac{\sqrt{233}+11}{2}$$

Now solve the equation p=\frac{11±\sqrt{233}}{2} when ± is minus. Subtract \sqrt{233} from 11.

$$p=\frac{11-\sqrt{233}}{2}$$

The equation is now solved.

$$p=\frac{\sqrt{233}+11}{2} p=\frac{11-\sqrt{233}}{2}$$

Steps for Completing the Square

Variable p cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by p+2.

$$\left(p+2\right)p-2=13\left(p+2\right)$$

Use the distributive property to multiply p+2 by p.

$$p^{2}+2p-2=13\left(p+2\right)$$

Use the distributive property to multiply 13 by p+2.

$$p^{2}+2p-2=13p+26$$

Subtract 13p from both sides.

$$p^{2}+2p-2-13p=26$$

Combine 2p and -13p to get -11p.

$$p^{2}-11p-2=26$$

Add 2 to both sides.

$$p^{2}-11p=26+2$$

Add 26 and 2 to get 28.

$$p^{2}-11p=28$$

Divide -11, the coefficient of the x term, by 2 to get -\frac{11}{2}. Then add the square of -\frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}-11p+\left(-\frac{11}{2}\right)^{2}=28+\left(-\frac{11}{2}\right)^{2}$$

Square -\frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$p^{2}-11p+\frac{121}{4}=28+\frac{121}{4}$$

Add 28 to \frac{121}{4}.

$$p^{2}-11p+\frac{121}{4}=\frac{233}{4}$$

Factor p^{2}-11p+\frac{121}{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}.

$$\left(p-\frac{11}{2}\right)^{2}=\frac{233}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p-\frac{11}{2}\right)^{2}}=\sqrt{\frac{233}{4}}$$

Simplify.

$$p-\frac{11}{2}=\frac{\sqrt{233}}{2} p-\frac{11}{2}=-\frac{\sqrt{233}}{2}$$

Add \frac{11}{2} to both sides of the equation.

$$p=\frac{\sqrt{233}+11}{2} p=\frac{11-\sqrt{233}}{2}$$