Solve v/v-3-26/v^2-9=7/v+3

$$\frac { v } { v - 3 } - \frac { 26 } { v ^ { 2 } - 9 } = \frac { 7 } { v + 3 }$$

Solve for v

$$v=-1$$

$$v=5$$

Steps Using Factoring

Variable v cannot be equal to any of the values -3,3 since division by zero is not defined. Multiply both sides of the equation by \left(v-3\right)\left(v+3\right), the least common multiple of v-3,v^{2}-9,v+3.

$$\left(v+3\right)v-26=\left(v-3\right)\times 7$$

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

$$v^{2}+3v-26=\left(v-3\right)\times 7$$

Use the distributive property to multiply v-3 by 7.

$$v^{2}+3v-26=7v-21$$

Subtract 7v from both sides.

$$v^{2}+3v-26-7v=-21$$

Combine 3v and -7v to get -4v.

$$v^{2}-4v-26=-21$$

Add 21 to both sides.

$$v^{2}-4v-26+21=0$$

Add -26 and 21 to get -5.

$$v^{2}-4v-5=0$$

To solve the equation, factor v^{2}-4v-5 using formula v^{2}+\left(a+b\right)v+ab=\left(v+a\right)\left(v+b\right). To find a and b, set up a system to be solved.

$$a+b=-4 ab=-5$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.

$$a=-5 b=1$$

Rewrite factored expression \left(v+a\right)\left(v+b\right) using the obtained values.

$$\left(v-5\right)\left(v+1\right)$$

To find equation solutions, solve v-5=0 and v+1=0.

$$v=5 v=-1$$

Steps Using Factoring By Grouping

Variable v cannot be equal to any of the values -3,3 since division by zero is not defined. Multiply both sides of the equation by \left(v-3\right)\left(v+3\right), the least common multiple of v-3,v^{2}-9,v+3.

$$\left(v+3\right)v-26=\left(v-3\right)\times 7$$

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

$$v^{2}+3v-26=\left(v-3\right)\times 7$$

Use the distributive property to multiply v-3 by 7.

$$v^{2}+3v-26=7v-21$$

Subtract 7v from both sides.

$$v^{2}+3v-26-7v=-21$$

Combine 3v and -7v to get -4v.

$$v^{2}-4v-26=-21$$

Add 21 to both sides.

$$v^{2}-4v-26+21=0$$

Add -26 and 21 to get -5.

$$v^{2}-4v-5=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as v^{2}+av+bv-5. To find a and b, set up a system to be solved.

$$a+b=-4 ab=1\left(-5\right)=-5$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.

$$a=-5 b=1$$

Rewrite v^{2}-4v-5 as \left(v^{2}-5v\right)+\left(v-5\right).

$$\left(v^{2}-5v\right)+\left(v-5\right)$$

Factor out v in v^{2}-5v.

$$v\left(v-5\right)+v-5$$

Factor out common term v-5 by using distributive property.

$$\left(v-5\right)\left(v+1\right)$$

To find equation solutions, solve v-5=0 and v+1=0.

$$v=5 v=-1$$

Steps Using the Quadratic Formula

Variable v cannot be equal to any of the values -3,3 since division by zero is not defined. Multiply both sides of the equation by \left(v-3\right)\left(v+3\right), the least common multiple of v-3,v^{2}-9,v+3.

$$\left(v+3\right)v-26=\left(v-3\right)\times 7$$

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

$$v^{2}+3v-26=\left(v-3\right)\times 7$$

Use the distributive property to multiply v-3 by 7.

$$v^{2}+3v-26=7v-21$$

Subtract 7v from both sides.

$$v^{2}+3v-26-7v=-21$$

Combine 3v and -7v to get -4v.

$$v^{2}-4v-26=-21$$

Add 21 to both sides.

$$v^{2}-4v-26+21=0$$

Add -26 and 21 to get -5.

$$v^{2}-4v-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-5\right)}}{2}$$

Square -4.

$$v=\frac{-\left(-4\right)±\sqrt{16-4\left(-5\right)}}{2}$$

Multiply -4 times -5.

$$v=\frac{-\left(-4\right)±\sqrt{16+20}}{2}$$

Add 16 to 20.

$$v=\frac{-\left(-4\right)±\sqrt{36}}{2}$$

Take the square root of 36.

$$v=\frac{-\left(-4\right)±6}{2}$$

The opposite of -4 is 4.

$$v=\frac{4±6}{2}$$

Now solve the equation v=\frac{4±6}{2} when ± is plus. Add 4 to 6.

$$v=\frac{10}{2}$$

Divide 10 by 2.

$$v=5$$

Now solve the equation v=\frac{4±6}{2} when ± is minus. Subtract 6 from 4.

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

Divide -2 by 2.

$$v=-1$$

The equation is now solved.

$$v=5 v=-1$$

Steps for Completing the Square

Variable v cannot be equal to any of the values -3,3 since division by zero is not defined. Multiply both sides of the equation by \left(v-3\right)\left(v+3\right), the least common multiple of v-3,v^{2}-9,v+3.

$$\left(v+3\right)v-26=\left(v-3\right)\times 7$$

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

$$v^{2}+3v-26=\left(v-3\right)\times 7$$

Use the distributive property to multiply v-3 by 7.

$$v^{2}+3v-26=7v-21$$

Subtract 7v from both sides.

$$v^{2}+3v-26-7v=-21$$

Combine 3v and -7v to get -4v.

$$v^{2}-4v-26=-21$$

Add 26 to both sides.

$$v^{2}-4v=-21+26$$

Add -21 and 26 to get 5.

$$v^{2}-4v=5$$

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

$$v^{2}-4v+\left(-2\right)^{2}=5+\left(-2\right)^{2}$$

Square -2.

$$v^{2}-4v+4=5+4$$

Add 5 to 4.

$$v^{2}-4v+4=9$$

Factor v^{2}-4v+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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$v-2=3 v-2=-3$$

Add 2 to both sides of the equation.

$$v=5 v=-1$$