Solve v-24/v=2

$$v - \frac { 24 } { v } = 2$$

Solve for v

$$v=-4$$

$$v=6$$

Steps Using Factoring

Variable v cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by v.

$$vv-24=2v$$

Multiply v and v to get v^{2}.

$$v^{2}-24=2v$$

Subtract 2v from both sides.

$$v^{2}-24-2v=0$$

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

$$v^{2}-2v-24=0$$

To solve the equation, factor v^{2}-2v-24 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.

$$a+b=-2 ab=-24$$

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. List all such integer pairs that give product -24.

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

Calculate the sum for each pair.

$$1-24=-23 2-12=-10 3-8=-5 4-6=-2$$

The solution is the pair that gives sum -2.

$$a=-6 b=4$$

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

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

To find equation solutions, solve v-6=0 and v+4=0.

$$v=6 v=-4$$

Steps Using Factoring By Grouping

Variable v cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by v.

$$vv-24=2v$$

Multiply v and v to get v^{2}.

$$v^{2}-24=2v$$

Subtract 2v from both sides.

$$v^{2}-24-2v=0$$

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

$$v^{2}-2v-24=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as v^{2}+av+bv-24. To find a and b, set up a system to be solved.

$$a+b=-2 ab=1\left(-24\right)=-24$$

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. List all such integer pairs that give product -24.

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

Calculate the sum for each pair.

$$1-24=-23 2-12=-10 3-8=-5 4-6=-2$$

The solution is the pair that gives sum -2.

$$a=-6 b=4$$

Rewrite v^{2}-2v-24 as \left(v^{2}-6v\right)+\left(4v-24\right).

$$\left(v^{2}-6v\right)+\left(4v-24\right)$$

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

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

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

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

To find equation solutions, solve v-6=0 and v+4=0.

$$v=6 v=-4$$

Steps Using the Quadratic Formula

Variable v cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by v.

$$vv-24=2v$$

Multiply v and v to get v^{2}.

$$v^{2}-24=2v$$

Subtract 2v from both sides.

$$v^{2}-24-2v=0$$

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.

$$v^{2}-2v-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-24\right)}}{2}$$

Square -2.

$$v=\frac{-\left(-2\right)±\sqrt{4-4\left(-24\right)}}{2}$$

Multiply -4 times -24.

$$v=\frac{-\left(-2\right)±\sqrt{4+96}}{2}$$

Add 4 to 96.

$$v=\frac{-\left(-2\right)±\sqrt{100}}{2}$$

Take the square root of 100.

$$v=\frac{-\left(-2\right)±10}{2}$$

The opposite of -2 is 2.

$$v=\frac{2±10}{2}$$

Now solve the equation v=\frac{2±10}{2} when ± is plus. Add 2 to 10.

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

Divide 12 by 2.

$$v=6$$

Now solve the equation v=\frac{2±10}{2} when ± is minus. Subtract 10 from 2.

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

Divide -8 by 2.

$$v=-4$$

The equation is now solved.

$$v=6 v=-4$$

Steps for Completing the Square

Variable v cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by v.

$$vv-24=2v$$

Multiply v and v to get v^{2}.

$$v^{2}-24=2v$$

Subtract 2v from both sides.

$$v^{2}-24-2v=0$$

Add 24 to both sides. Anything plus zero gives itself.

$$v^{2}-2v=24$$

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.

$$v^{2}-2v+1=24+1$$

Add 24 to 1.

$$v^{2}-2v+1=25$$

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}.

$$\left(v-1\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v-1\right)^{2}}=\sqrt{25}$$

Simplify.

$$v-1=5 v-1=-5$$

Add 1 to both sides of the equation.

$$v=6 v=-4$$