Solve -8v/v^2-2v-15-4/v-5=v/v+3

$$\frac { - 8 v } { v ^ { 2 } - 2 v - 15 } - \frac { 4 } { v - 5 } = \frac { v } { v + 3 }$$

Solve for v

$$v=-4$$

Steps Using Factoring By Grouping

Variable v cannot be equal to any of the values -3,5 since division by zero is not defined. Multiply both sides of the equation by \left(v-5\right)\left(v+3\right), the least common multiple of v^{2}-2v-15,v-5,v+3.

$$-8v-\left(v+3\right)\times 4=\left(v-5\right)v$$

Use the distributive property to multiply v+3 by 4.

$$-8v-\left(4v+12\right)=\left(v-5\right)v$$

To find the opposite of 4v+12, find the opposite of each term.

$$-8v-4v-12=\left(v-5\right)v$$

Combine -8v and -4v to get -12v.

$$-12v-12=\left(v-5\right)v$$

Use the distributive property to multiply v-5 by v.

$$-12v-12=v^{2}-5v$$

Subtract v^{2} from both sides.

$$-12v-12-v^{2}=-5v$$

Add 5v to both sides.

$$-12v-12-v^{2}+5v=0$$

Combine -12v and 5v to get -7v.

$$-7v-12-v^{2}=0$$

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

$$-v^{2}-7v-12=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -v^{2}+av+bv-12. To find a and b, set up a system to be solved.

$$a+b=-7 ab=-\left(-12\right)=12$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 12.

$$-1,-12 -2,-6 -3,-4$$

Calculate the sum for each pair.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

The solution is the pair that gives sum -7.

$$a=-3 b=-4$$

Rewrite -v^{2}-7v-12 as \left(-v^{2}-3v\right)+\left(-4v-12\right).

$$\left(-v^{2}-3v\right)+\left(-4v-12\right)$$

Factor out v in the first and 4 in the second group.

$$v\left(-v-3\right)+4\left(-v-3\right)$$

Factor out common term -v-3 by using distributive property.

$$\left(-v-3\right)\left(v+4\right)$$

To find equation solutions, solve -v-3=0 and v+4=0.

$$v=-3 v=-4$$

Variable v cannot be equal to -3.

$$v=-4$$

Steps Using the Quadratic Formula

Variable v cannot be equal to any of the values -3,5 since division by zero is not defined. Multiply both sides of the equation by \left(v-5\right)\left(v+3\right), the least common multiple of v^{2}-2v-15,v-5,v+3.

$$-8v-\left(v+3\right)\times 4=\left(v-5\right)v$$

Use the distributive property to multiply v+3 by 4.

$$-8v-\left(4v+12\right)=\left(v-5\right)v$$

To find the opposite of 4v+12, find the opposite of each term.

$$-8v-4v-12=\left(v-5\right)v$$

Combine -8v and -4v to get -12v.

$$-12v-12=\left(v-5\right)v$$

Use the distributive property to multiply v-5 by v.

$$-12v-12=v^{2}-5v$$

Subtract v^{2} from both sides.

$$-12v-12-v^{2}=-5v$$

Add 5v to both sides.

$$-12v-12-v^{2}+5v=0$$

Combine -12v and 5v to get -7v.

$$-7v-12-v^{2}=0$$

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.

$$-v^{2}-7v-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -7 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-1\right)\left(-12\right)}}{2\left(-1\right)}$$

Square -7.

$$v=\frac{-\left(-7\right)±\sqrt{49-4\left(-1\right)\left(-12\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$v=\frac{-\left(-7\right)±\sqrt{49+4\left(-12\right)}}{2\left(-1\right)}$$

Multiply 4 times -12.

$$v=\frac{-\left(-7\right)±\sqrt{49-48}}{2\left(-1\right)}$$

Add 49 to -48.

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

Take the square root of 1.

$$v=\frac{-\left(-7\right)±1}{2\left(-1\right)}$$

The opposite of -7 is 7.

$$v=\frac{7±1}{2\left(-1\right)}$$

Multiply 2 times -1.

$$v=\frac{7±1}{-2}$$

Now solve the equation v=\frac{7±1}{-2} when ± is plus. Add 7 to 1.

$$v=\frac{8}{-2}$$

Divide 8 by -2.

$$v=-4$$

Now solve the equation v=\frac{7±1}{-2} when ± is minus. Subtract 1 from 7.

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

Divide 6 by -2.

$$v=-3$$

The equation is now solved.

$$v=-4 v=-3$$

Variable v cannot be equal to -3.

$$v=-4$$

Steps for Completing the Square

Variable v cannot be equal to any of the values -3,5 since division by zero is not defined. Multiply both sides of the equation by \left(v-5\right)\left(v+3\right), the least common multiple of v^{2}-2v-15,v-5,v+3.

$$-8v-\left(v+3\right)\times 4=\left(v-5\right)v$$

Use the distributive property to multiply v+3 by 4.

$$-8v-\left(4v+12\right)=\left(v-5\right)v$$

To find the opposite of 4v+12, find the opposite of each term.

$$-8v-4v-12=\left(v-5\right)v$$

Combine -8v and -4v to get -12v.

$$-12v-12=\left(v-5\right)v$$

Use the distributive property to multiply v-5 by v.

$$-12v-12=v^{2}-5v$$

Subtract v^{2} from both sides.

$$-12v-12-v^{2}=-5v$$

Add 5v to both sides.

$$-12v-12-v^{2}+5v=0$$

Combine -12v and 5v to get -7v.

$$-7v-12-v^{2}=0$$

Add 12 to both sides. Anything plus zero gives itself.

$$-7v-v^{2}=12$$

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.

$$-v^{2}-7v=12$$

Divide both sides by -1.

$$\frac{-v^{2}-7v}{-1}=\frac{12}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$v^{2}+\left(-\frac{7}{-1}\right)v=\frac{12}{-1}$$

Divide -7 by -1.

$$v^{2}+7v=\frac{12}{-1}$$

Divide 12 by -1.

$$v^{2}+7v=-12$$

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.

$$v^{2}+7v+\left(\frac{7}{2}\right)^{2}=-12+\left(\frac{7}{2}\right)^{2}$$

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

$$v^{2}+7v+\frac{49}{4}=-12+\frac{49}{4}$$

Add -12 to \frac{49}{4}.

$$v^{2}+7v+\frac{49}{4}=\frac{1}{4}$$

Factor v^{2}+7v+\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}.

$$\left(v+\frac{7}{2}\right)^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v+\frac{7}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

$$v+\frac{7}{2}=\frac{1}{2} v+\frac{7}{2}=-\frac{1}{2}$$

Subtract \frac{7}{2} from both sides of the equation.

$$v=-3 v=-4$$

Variable v cannot be equal to -3.

$$v=-4$$