Solve 2/x=x/5x+12

$$\frac { 2 } { x } = \frac { x } { 5 x + 12 }$$

Solve for x

$$x=-2$$

$$x=12$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -\frac{12}{5},0 since division by zero is not defined. Multiply both sides of the equation by x\left(5x+12\right), the least common multiple of x,5x+12.

$$\left(5x+12\right)\times 2=xx$$

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

$$\left(5x+12\right)\times 2=x^{2}$$

Use the distributive property to multiply 5x+12 by 2.

$$10x+24=x^{2}$$

Subtract x^{2} from both sides.

$$10x+24-x^{2}=0$$

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

$$-x^{2}+10x+24=0$$

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

$$a+b=10 ab=-24=-24$$

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 -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 10.

$$a=12 b=-2$$

Rewrite -x^{2}+10x+24 as \left(-x^{2}+12x\right)+\left(-2x+24\right).

$$\left(-x^{2}+12x\right)+\left(-2x+24\right)$$

Factor out -x in the first and -2 in the second group.

$$-x\left(x-12\right)-2\left(x-12\right)$$

Factor out common term x-12 by using distributive property.

$$\left(x-12\right)\left(-x-2\right)$$

To find equation solutions, solve x-12=0 and -x-2=0.

$$x=12 x=-2$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -\frac{12}{5},0 since division by zero is not defined. Multiply both sides of the equation by x\left(5x+12\right), the least common multiple of x,5x+12.

$$\left(5x+12\right)\times 2=xx$$

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

$$\left(5x+12\right)\times 2=x^{2}$$

Use the distributive property to multiply 5x+12 by 2.

$$10x+24=x^{2}$$

Subtract x^{2} from both sides.

$$10x+24-x^{2}=0$$

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.

$$-x^{2}+10x+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 10 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-1\right)\times 24}}{2\left(-1\right)}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\left(-1\right)\times 24}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-10±\sqrt{100+4\times 24}}{2\left(-1\right)}$$

Multiply 4 times 24.

$$x=\frac{-10±\sqrt{100+96}}{2\left(-1\right)}$$

Add 100 to 96.

$$x=\frac{-10±\sqrt{196}}{2\left(-1\right)}$$

Take the square root of 196.

$$x=\frac{-10±14}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-10±14}{-2}$$

Now solve the equation x=\frac{-10±14}{-2} when ± is plus. Add -10 to 14.

$$x=\frac{4}{-2}$$

Divide 4 by -2.

$$x=-2$$

Now solve the equation x=\frac{-10±14}{-2} when ± is minus. Subtract 14 from -10.

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

Divide -24 by -2.

$$x=12$$

The equation is now solved.

$$x=-2 x=12$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -\frac{12}{5},0 since division by zero is not defined. Multiply both sides of the equation by x\left(5x+12\right), the least common multiple of x,5x+12.

$$\left(5x+12\right)\times 2=xx$$

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

$$\left(5x+12\right)\times 2=x^{2}$$

Use the distributive property to multiply 5x+12 by 2.

$$10x+24=x^{2}$$

Subtract x^{2} from both sides.

$$10x+24-x^{2}=0$$

Subtract 24 from both sides. Anything subtracted from zero gives its negation.

$$10x-x^{2}=-24$$

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.

$$-x^{2}+10x=-24$$

Divide both sides by -1.

$$\frac{-x^{2}+10x}{-1}=-\frac{24}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{10}{-1}x=-\frac{24}{-1}$$

Divide 10 by -1.

$$x^{2}-10x=-\frac{24}{-1}$$

Divide -24 by -1.

$$x^{2}-10x=24$$

Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-10x+\left(-5\right)^{2}=24+\left(-5\right)^{2}$$

Square -5.

$$x^{2}-10x+25=24+25$$

Add 24 to 25.

$$x^{2}-10x+25=49$$

Factor x^{2}-10x+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}.

$$\left(x-5\right)^{2}=49$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{49}$$

Simplify.

$$x-5=7 x-5=-7$$

Add 5 to both sides of the equation.

$$x=12 x=-2$$