Solve x+1/3x=18/2x+32

$$\frac { x + 1 } { 3 x } = \frac { 18 } { 2 x + 32 }$$

Solve for x

$$x=2$$

$$x=8$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -16,0 since division by zero is not defined. Multiply both sides of the equation by 6x\left(x+16\right), the least common multiple of 3x,2x+32.

$$\left(2x+32\right)\left(x+1\right)=3x\times 18$$

Use the distributive property to multiply 2x+32 by x+1 and combine like terms.

$$2x^{2}+34x+32=3x\times 18$$

Multiply 3 and 18 to get 54.

$$2x^{2}+34x+32=54x$$

Subtract 54x from both sides.

$$2x^{2}+34x+32-54x=0$$

Combine 34x and -54x to get -20x.

$$2x^{2}-20x+32=0$$

Divide both sides by 2.

$$x^{2}-10x+16=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+16. To find a and b, set up a system to be solved.

$$a+b=-10 ab=1\times 16=16$$

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 16.

$$-1,-16 -2,-8 -4,-4$$

Calculate the sum for each pair.

$$-1-16=-17 -2-8=-10 -4-4=-8$$

The solution is the pair that gives sum -10.

$$a=-8 b=-2$$

Rewrite x^{2}-10x+16 as \left(x^{2}-8x\right)+\left(-2x+16\right).

$$\left(x^{2}-8x\right)+\left(-2x+16\right)$$

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

$$x\left(x-8\right)-2\left(x-8\right)$$

Factor out common term x-8 by using distributive property.

$$\left(x-8\right)\left(x-2\right)$$

To find equation solutions, solve x-8=0 and x-2=0.

$$x=8 x=2$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -16,0 since division by zero is not defined. Multiply both sides of the equation by 6x\left(x+16\right), the least common multiple of 3x,2x+32.

$$\left(2x+32\right)\left(x+1\right)=3x\times 18$$

Use the distributive property to multiply 2x+32 by x+1 and combine like terms.

$$2x^{2}+34x+32=3x\times 18$$

Multiply 3 and 18 to get 54.

$$2x^{2}+34x+32=54x$$

Subtract 54x from both sides.

$$2x^{2}+34x+32-54x=0$$

Combine 34x and -54x to get -20x.

$$2x^{2}-20x+32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -20 for b, and 32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 2\times 32}}{2\times 2}$$

Square -20.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\times 2\times 32}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-20\right)±\sqrt{400-8\times 32}}{2\times 2}$$

Multiply -8 times 32.

$$x=\frac{-\left(-20\right)±\sqrt{400-256}}{2\times 2}$$

Add 400 to -256.

$$x=\frac{-\left(-20\right)±\sqrt{144}}{2\times 2}$$

Take the square root of 144.

$$x=\frac{-\left(-20\right)±12}{2\times 2}$$

The opposite of -20 is 20.

$$x=\frac{20±12}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{20±12}{4}$$

Now solve the equation x=\frac{20±12}{4} when ± is plus. Add 20 to 12.

$$x=\frac{32}{4}$$

Divide 32 by 4.

$$x=8$$

Now solve the equation x=\frac{20±12}{4} when ± is minus. Subtract 12 from 20.

$$x=\frac{8}{4}$$

Divide 8 by 4.

$$x=2$$

The equation is now solved.

$$x=8 x=2$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -16,0 since division by zero is not defined. Multiply both sides of the equation by 6x\left(x+16\right), the least common multiple of 3x,2x+32.

$$\left(2x+32\right)\left(x+1\right)=3x\times 18$$

Use the distributive property to multiply 2x+32 by x+1 and combine like terms.

$$2x^{2}+34x+32=3x\times 18$$

Multiply 3 and 18 to get 54.

$$2x^{2}+34x+32=54x$$

Subtract 54x from both sides.

$$2x^{2}+34x+32-54x=0$$

Combine 34x and -54x to get -20x.

$$2x^{2}-20x+32=0$$

Subtract 32 from both sides. Anything subtracted from zero gives its negation.

$$2x^{2}-20x=-32$$

Divide both sides by 2.

$$\frac{2x^{2}-20x}{2}=-\frac{32}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{20}{2}\right)x=-\frac{32}{2}$$

Divide -20 by 2.

$$x^{2}-10x=-\frac{32}{2}$$

Divide -32 by 2.

$$x^{2}-10x=-16$$

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.

$$x^{2}-10x+\left(-5\right)^{2}=-16+\left(-5\right)^{2}$$

Square -5.

$$x^{2}-10x+25=-16+25$$

Add -16 to 25.

$$x^{2}-10x+25=9$$

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}.

$$\left(x-5\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{9}$$

Simplify.

$$x-5=3 x-5=-3$$

Add 5 to both sides of the equation.

$$x=8 x=2$$