Solve 1/p+5p-5/p^2-2p=1

$$\frac { 1 } { p } + \frac { 5 p - 5 } { p ^ { 2 } - 2 p } = 1$$

Solve for p

$$p=1$$

$$p=7$$

Steps Using Factoring By Grouping

Variable p cannot be equal to any of the values 0,2 since division by zero is not defined. Multiply both sides of the equation by p\left(p-2\right), the least common multiple of p,p^{2}-2p.

$$p-2+5p-5=p\left(p-2\right)$$

Combine p and 5p to get 6p.

$$6p-2-5=p\left(p-2\right)$$

Subtract 5 from -2 to get -7.

$$6p-7=p\left(p-2\right)$$

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

$$6p-7=p^{2}-2p$$

Subtract p^{2} from both sides.

$$6p-7-p^{2}=-2p$$

Add 2p to both sides.

$$6p-7-p^{2}+2p=0$$

Combine 6p and 2p to get 8p.

$$8p-7-p^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-p^{2}+8p-7=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -p^{2}+ap+bp-7. To find a and b, set up a system to be solved.

$$a+b=8 ab=-\left(-7\right)=7$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=7 b=1$$

Rewrite -p^{2}+8p-7 as \left(-p^{2}+7p\right)+\left(p-7\right).

$$\left(-p^{2}+7p\right)+\left(p-7\right)$$

Factor out -p in -p^{2}+7p.

$$-p\left(p-7\right)+p-7$$

Factor out common term p-7 by using distributive property.

$$\left(p-7\right)\left(-p+1\right)$$

To find equation solutions, solve p-7=0 and -p+1=0.

$$p=7 p=1$$

Steps Using the Quadratic Formula

Variable p cannot be equal to any of the values 0,2 since division by zero is not defined. Multiply both sides of the equation by p\left(p-2\right), the least common multiple of p,p^{2}-2p.

$$p-2+5p-5=p\left(p-2\right)$$

Combine p and 5p to get 6p.

$$6p-2-5=p\left(p-2\right)$$

Subtract 5 from -2 to get -7.

$$6p-7=p\left(p-2\right)$$

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

$$6p-7=p^{2}-2p$$

Subtract p^{2} from both sides.

$$6p-7-p^{2}=-2p$$

Add 2p to both sides.

$$6p-7-p^{2}+2p=0$$

Combine 6p and 2p to get 8p.

$$8p-7-p^{2}=0$$

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.

$$-p^{2}+8p-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 8 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-8±\sqrt{8^{2}-4\left(-1\right)\left(-7\right)}}{2\left(-1\right)}$$

Square 8.

$$p=\frac{-8±\sqrt{64-4\left(-1\right)\left(-7\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$p=\frac{-8±\sqrt{64+4\left(-7\right)}}{2\left(-1\right)}$$

Multiply 4 times -7.

$$p=\frac{-8±\sqrt{64-28}}{2\left(-1\right)}$$

Add 64 to -28.

$$p=\frac{-8±\sqrt{36}}{2\left(-1\right)}$$

Take the square root of 36.

$$p=\frac{-8±6}{2\left(-1\right)}$$

Multiply 2 times -1.

$$p=\frac{-8±6}{-2}$$

Now solve the equation p=\frac{-8±6}{-2} when ± is plus. Add -8 to 6.

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

Divide -2 by -2.

$$p=1$$

Now solve the equation p=\frac{-8±6}{-2} when ± is minus. Subtract 6 from -8.

$$p=-\frac{14}{-2}$$

Divide -14 by -2.

$$p=7$$

The equation is now solved.

$$p=1 p=7$$

Steps for Completing the Square

Variable p cannot be equal to any of the values 0,2 since division by zero is not defined. Multiply both sides of the equation by p\left(p-2\right), the least common multiple of p,p^{2}-2p.

$$p-2+5p-5=p\left(p-2\right)$$

Combine p and 5p to get 6p.

$$6p-2-5=p\left(p-2\right)$$

Subtract 5 from -2 to get -7.

$$6p-7=p\left(p-2\right)$$

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

$$6p-7=p^{2}-2p$$

Subtract p^{2} from both sides.

$$6p-7-p^{2}=-2p$$

Add 2p to both sides.

$$6p-7-p^{2}+2p=0$$

Combine 6p and 2p to get 8p.

$$8p-7-p^{2}=0$$

Add 7 to both sides. Anything plus zero gives itself.

$$8p-p^{2}=7$$

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.

$$-p^{2}+8p=7$$

Divide both sides by -1.

$$\frac{-p^{2}+8p}{-1}=\frac{7}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$p^{2}+\frac{8}{-1}p=\frac{7}{-1}$$

Divide 8 by -1.

$$p^{2}-8p=\frac{7}{-1}$$

Divide 7 by -1.

$$p^{2}-8p=-7$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}-8p+\left(-4\right)^{2}=-7+\left(-4\right)^{2}$$

Square -4.

$$p^{2}-8p+16=-7+16$$

Add -7 to 16.

$$p^{2}-8p+16=9$$

Factor p^{2}-8p+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}.

$$\left(p-4\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p-4\right)^{2}}=\sqrt{9}$$

Simplify.

$$p-4=3 p-4=-3$$

Add 4 to both sides of the equation.

$$p=7 p=1$$