Solve 6/v^2-4v=1/8-v

$$\frac { 6 } { v ^ { 2 } - 4 v } = \frac { 1 } { 8 - v }$$

Solve for v

$$v=-8$$

$$v=6$$

Steps Using Factoring

Variable v cannot be equal to any of the values 0,4,8 since division by zero is not defined. Multiply both sides of the equation by v\left(v-8\right)\left(v-4\right), the least common multiple of v^{2}-4v,8-v.

$$\left(v-8\right)\times 6=-v\left(-4+v\right)$$

Use the distributive property to multiply v-8 by 6.

$$6v-48=-v\left(-4+v\right)$$

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

$$6v-48=4v-v^{2}$$

Subtract 4v from both sides.

$$6v-48-4v=-v^{2}$$

Combine 6v and -4v to get 2v.

$$2v-48=-v^{2}$$

Add v^{2} to both sides.

$$2v-48+v^{2}=0$$

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

$$v^{2}+2v-48=0$$

To solve the equation, factor v^{2}+2v-48 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.

$$a+b=2 ab=-48$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -48.

$$-1,48 -2,24 -3,16 -4,12 -6,8$$

Calculate the sum for each pair.

$$-1+48=47 -2+24=22 -3+16=13 -4+12=8 -6+8=2$$

The solution is the pair that gives sum 2.

$$a=-6 b=8$$

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

$$\left(v-6\right)\left(v+8\right)$$

To find equation solutions, solve v-6=0 and v+8=0.

$$v=6 v=-8$$

Steps Using Factoring By Grouping

Variable v cannot be equal to any of the values 0,4,8 since division by zero is not defined. Multiply both sides of the equation by v\left(v-8\right)\left(v-4\right), the least common multiple of v^{2}-4v,8-v.

$$\left(v-8\right)\times 6=-v\left(-4+v\right)$$

Use the distributive property to multiply v-8 by 6.

$$6v-48=-v\left(-4+v\right)$$

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

$$6v-48=4v-v^{2}$$

Subtract 4v from both sides.

$$6v-48-4v=-v^{2}$$

Combine 6v and -4v to get 2v.

$$2v-48=-v^{2}$$

Add v^{2} to both sides.

$$2v-48+v^{2}=0$$

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

$$v^{2}+2v-48=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as v^{2}+av+bv-48. To find a and b, set up a system to be solved.

$$a+b=2 ab=1\left(-48\right)=-48$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -48.

$$-1,48 -2,24 -3,16 -4,12 -6,8$$

Calculate the sum for each pair.

$$-1+48=47 -2+24=22 -3+16=13 -4+12=8 -6+8=2$$

The solution is the pair that gives sum 2.

$$a=-6 b=8$$

Rewrite v^{2}+2v-48 as \left(v^{2}-6v\right)+\left(8v-48\right).

$$\left(v^{2}-6v\right)+\left(8v-48\right)$$

Factor out v in the first and 8 in the second group.

$$v\left(v-6\right)+8\left(v-6\right)$$

Factor out common term v-6 by using distributive property.

$$\left(v-6\right)\left(v+8\right)$$

To find equation solutions, solve v-6=0 and v+8=0.

$$v=6 v=-8$$

Steps Using the Quadratic Formula

Variable v cannot be equal to any of the values 0,4,8 since division by zero is not defined. Multiply both sides of the equation by v\left(v-8\right)\left(v-4\right), the least common multiple of v^{2}-4v,8-v.

$$\left(v-8\right)\times 6=-v\left(-4+v\right)$$

Use the distributive property to multiply v-8 by 6.

$$6v-48=-v\left(-4+v\right)$$

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

$$6v-48=4v-v^{2}$$

Subtract 4v from both sides.

$$6v-48-4v=-v^{2}$$

Combine 6v and -4v to get 2v.

$$2v-48=-v^{2}$$

Add v^{2} to both sides.

$$2v-48+v^{2}=0$$

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.

$$v^{2}+2v-48=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-2±\sqrt{2^{2}-4\left(-48\right)}}{2}$$

Square 2.

$$v=\frac{-2±\sqrt{4-4\left(-48\right)}}{2}$$

Multiply -4 times -48.

$$v=\frac{-2±\sqrt{4+192}}{2}$$

Add 4 to 192.

$$v=\frac{-2±\sqrt{196}}{2}$$

Take the square root of 196.

$$v=\frac{-2±14}{2}$$

Now solve the equation v=\frac{-2±14}{2} when ± is plus. Add -2 to 14.

$$v=\frac{12}{2}$$

Divide 12 by 2.

$$v=6$$

Now solve the equation v=\frac{-2±14}{2} when ± is minus. Subtract 14 from -2.

$$v=-\frac{16}{2}$$

Divide -16 by 2.

$$v=-8$$

The equation is now solved.

$$v=6 v=-8$$

Steps for Completing the Square

Variable v cannot be equal to any of the values 0,4,8 since division by zero is not defined. Multiply both sides of the equation by v\left(v-8\right)\left(v-4\right), the least common multiple of v^{2}-4v,8-v.

$$\left(v-8\right)\times 6=-v\left(-4+v\right)$$

Use the distributive property to multiply v-8 by 6.

$$6v-48=-v\left(-4+v\right)$$

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

$$6v-48=4v-v^{2}$$

Subtract 4v from both sides.

$$6v-48-4v=-v^{2}$$

Combine 6v and -4v to get 2v.

$$2v-48=-v^{2}$$

Add v^{2} to both sides.

$$2v-48+v^{2}=0$$

Add 48 to both sides. Anything plus zero gives itself.

$$2v+v^{2}=48$$

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.

$$v^{2}+2v=48$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$v^{2}+2v+1^{2}=48+1^{2}$$

Square 1.

$$v^{2}+2v+1=48+1$$

Add 48 to 1.

$$v^{2}+2v+1=49$$

Factor v^{2}+2v+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(v+1\right)^{2}=49$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v+1\right)^{2}}=\sqrt{49}$$

Simplify.

$$v+1=7 v+1=-7$$

Subtract 1 from both sides of the equation.

$$v=6 v=-8$$