Solve 15/p+6p-5/p+2=1

$$\frac { 15 } { p } + \frac { 6 p - 5 } { p + 2 } = 1$$

Solve for p

$$p=\frac{-4+\sqrt{134}i}{5}\approx -0.8+2.315167381i$$

$$p=\frac{-\sqrt{134}i-4}{5}\approx -0.8-2.315167381i$$

Steps Using the Quadratic Formula

Variable p cannot be equal to any of the values -2,0 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.

$$\left(p+2\right)\times 15+p\left(6p-5\right)=p\left(p+2\right)$$

Use the distributive property to multiply p+2 by 15.

$$15p+30+p\left(6p-5\right)=p\left(p+2\right)$$

Use the distributive property to multiply p by 6p-5.

$$15p+30+6p^{2}-5p=p\left(p+2\right)$$

Combine 15p and -5p to get 10p.

$$10p+30+6p^{2}=p\left(p+2\right)$$

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

$$10p+30+6p^{2}=p^{2}+2p$$

Subtract p^{2} from both sides.

$$10p+30+6p^{2}-p^{2}=2p$$

Combine 6p^{2} and -p^{2} to get 5p^{2}.

$$10p+30+5p^{2}=2p$$

Subtract 2p from both sides.

$$10p+30+5p^{2}-2p=0$$

Combine 10p and -2p to get 8p.

$$8p+30+5p^{2}=0$$

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.

$$5p^{2}+8p+30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 8 for b, and 30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-8±\sqrt{8^{2}-4\times 5\times 30}}{2\times 5}$$

Square 8.

$$p=\frac{-8±\sqrt{64-4\times 5\times 30}}{2\times 5}$$

Multiply -4 times 5.

$$p=\frac{-8±\sqrt{64-20\times 30}}{2\times 5}$$

Multiply -20 times 30.

$$p=\frac{-8±\sqrt{64-600}}{2\times 5}$$

Add 64 to -600.

$$p=\frac{-8±\sqrt{-536}}{2\times 5}$$

Take the square root of -536.

$$p=\frac{-8±2\sqrt{134}i}{2\times 5}$$

Multiply 2 times 5.

$$p=\frac{-8±2\sqrt{134}i}{10}$$

Now solve the equation p=\frac{-8±2\sqrt{134}i}{10} when ± is plus. Add -8 to 2i\sqrt{134}.

$$p=\frac{-8+2\sqrt{134}i}{10}$$

Divide -8+2i\sqrt{134} by 10.

$$p=\frac{-4+\sqrt{134}i}{5}$$

Now solve the equation p=\frac{-8±2\sqrt{134}i}{10} when ± is minus. Subtract 2i\sqrt{134} from -8.

$$p=\frac{-2\sqrt{134}i-8}{10}$$

Divide -8-2i\sqrt{134} by 10.

$$p=\frac{-\sqrt{134}i-4}{5}$$

The equation is now solved.

$$p=\frac{-4+\sqrt{134}i}{5} p=\frac{-\sqrt{134}i-4}{5}$$

Steps for Completing the Square

Variable p cannot be equal to any of the values -2,0 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.

$$\left(p+2\right)\times 15+p\left(6p-5\right)=p\left(p+2\right)$$

Use the distributive property to multiply p+2 by 15.

$$15p+30+p\left(6p-5\right)=p\left(p+2\right)$$

Use the distributive property to multiply p by 6p-5.

$$15p+30+6p^{2}-5p=p\left(p+2\right)$$

Combine 15p and -5p to get 10p.

$$10p+30+6p^{2}=p\left(p+2\right)$$

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

$$10p+30+6p^{2}=p^{2}+2p$$

Subtract p^{2} from both sides.

$$10p+30+6p^{2}-p^{2}=2p$$

Combine 6p^{2} and -p^{2} to get 5p^{2}.

$$10p+30+5p^{2}=2p$$

Subtract 2p from both sides.

$$10p+30+5p^{2}-2p=0$$

Combine 10p and -2p to get 8p.

$$8p+30+5p^{2}=0$$

Subtract 30 from both sides. Anything subtracted from zero gives its negation.

$$8p+5p^{2}=-30$$

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.

$$5p^{2}+8p=-30$$

Divide both sides by 5.

$$\frac{5p^{2}+8p}{5}=-\frac{30}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$p^{2}+\frac{8}{5}p=-\frac{30}{5}$$

Divide -30 by 5.

$$p^{2}+\frac{8}{5}p=-6$$

Divide \frac{8}{5}, the coefficient of the x term, by 2 to get \frac{4}{5}. Then add the square of \frac{4}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}+\frac{8}{5}p+\left(\frac{4}{5}\right)^{2}=-6+\left(\frac{4}{5}\right)^{2}$$

Square \frac{4}{5} by squaring both the numerator and the denominator of the fraction.

$$p^{2}+\frac{8}{5}p+\frac{16}{25}=-6+\frac{16}{25}$$

Add -6 to \frac{16}{25}.

$$p^{2}+\frac{8}{5}p+\frac{16}{25}=-\frac{134}{25}$$

Factor p^{2}+\frac{8}{5}p+\frac{16}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(p+\frac{4}{5}\right)^{2}=-\frac{134}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p+\frac{4}{5}\right)^{2}}=\sqrt{-\frac{134}{25}}$$

Simplify.

$$p+\frac{4}{5}=\frac{\sqrt{134}i}{5} p+\frac{4}{5}=-\frac{\sqrt{134}i}{5}$$

Subtract \frac{4}{5} from both sides of the equation.

$$p=\frac{-4+\sqrt{134}i}{5} p=\frac{-\sqrt{134}i-4}{5}$$