Solve x+18/2+x/2+15=x^2+18x/2x+18

$$\frac{ x+18 }{ 2 } + \frac{ x }{ 2 } +15= \frac{ { x }^{ 2 } +18x }{ 2x+18 }$$

Solve for x

$$x=-36$$

$$x=-12$$

Steps Using Factoring

Variable x cannot be equal to -9 since division by zero is not defined. Multiply both sides of the equation by 2\left(x+9\right), the least common multiple of 2,2x+18.

$$\left(x+9\right)\left(x+18\right)+\left(x+9\right)x+2\left(x+9\right)\times 15=x^{2}+18x$$

Use the distributive property to multiply x+9 by x+18 and combine like terms.

$$x^{2}+27x+162+\left(x+9\right)x+2\left(x+9\right)\times 15=x^{2}+18x$$

Use the distributive property to multiply x+9 by x.

$$x^{2}+27x+162+x^{2}+9x+2\left(x+9\right)\times 15=x^{2}+18x$$

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

$$2x^{2}+27x+162+9x+2\left(x+9\right)\times 15=x^{2}+18x$$

Combine 27x and 9x to get 36x.

$$2x^{2}+36x+162+2\left(x+9\right)\times 15=x^{2}+18x$$

Multiply 2 and 15 to get 30.

$$2x^{2}+36x+162+30\left(x+9\right)=x^{2}+18x$$

Use the distributive property to multiply 30 by x+9.

$$2x^{2}+36x+162+30x+270=x^{2}+18x$$

Combine 36x and 30x to get 66x.

$$2x^{2}+66x+162+270=x^{2}+18x$$

Add 162 and 270 to get 432.

$$2x^{2}+66x+432=x^{2}+18x$$

Subtract x^{2} from both sides.

$$2x^{2}+66x+432-x^{2}=18x$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$x^{2}+66x+432=18x$$

Subtract 18x from both sides.

$$x^{2}+66x+432-18x=0$$

Combine 66x and -18x to get 48x.

$$x^{2}+48x+432=0$$

To solve the equation, factor x^{2}+48x+432 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=48 ab=432$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 432.

$$1,432 2,216 3,144 4,108 6,72 8,54 9,48 12,36 16,27 18,24$$

Calculate the sum for each pair.

$$1+432=433 2+216=218 3+144=147 4+108=112 6+72=78 8+54=62 9+48=57 12+36=48 16+27=43 18+24=42$$

The solution is the pair that gives sum 48.

$$a=12 b=36$$

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

$$\left(x+12\right)\left(x+36\right)$$

To find equation solutions, solve x+12=0 and x+36=0.

$$x=-12 x=-36$$

Steps Using Factoring By Grouping

Variable x cannot be equal to -9 since division by zero is not defined. Multiply both sides of the equation by 2\left(x+9\right), the least common multiple of 2,2x+18.

$$\left(x+9\right)\left(x+18\right)+\left(x+9\right)x+2\left(x+9\right)\times 15=x^{2}+18x$$

Use the distributive property to multiply x+9 by x+18 and combine like terms.

$$x^{2}+27x+162+\left(x+9\right)x+2\left(x+9\right)\times 15=x^{2}+18x$$

Use the distributive property to multiply x+9 by x.

$$x^{2}+27x+162+x^{2}+9x+2\left(x+9\right)\times 15=x^{2}+18x$$

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

$$2x^{2}+27x+162+9x+2\left(x+9\right)\times 15=x^{2}+18x$$

Combine 27x and 9x to get 36x.

$$2x^{2}+36x+162+2\left(x+9\right)\times 15=x^{2}+18x$$

Multiply 2 and 15 to get 30.

$$2x^{2}+36x+162+30\left(x+9\right)=x^{2}+18x$$

Use the distributive property to multiply 30 by x+9.

$$2x^{2}+36x+162+30x+270=x^{2}+18x$$

Combine 36x and 30x to get 66x.

$$2x^{2}+66x+162+270=x^{2}+18x$$

Add 162 and 270 to get 432.

$$2x^{2}+66x+432=x^{2}+18x$$

Subtract x^{2} from both sides.

$$2x^{2}+66x+432-x^{2}=18x$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$x^{2}+66x+432=18x$$

Subtract 18x from both sides.

$$x^{2}+66x+432-18x=0$$

Combine 66x and -18x to get 48x.

$$x^{2}+48x+432=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+432. To find a and b, set up a system to be solved.

$$a+b=48 ab=1\times 432=432$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 432.

$$1,432 2,216 3,144 4,108 6,72 8,54 9,48 12,36 16,27 18,24$$

Calculate the sum for each pair.

$$1+432=433 2+216=218 3+144=147 4+108=112 6+72=78 8+54=62 9+48=57 12+36=48 16+27=43 18+24=42$$

The solution is the pair that gives sum 48.

$$a=12 b=36$$

Rewrite x^{2}+48x+432 as \left(x^{2}+12x\right)+\left(36x+432\right).

$$\left(x^{2}+12x\right)+\left(36x+432\right)$$

Factor out x in the first and 36 in the second group.

$$x\left(x+12\right)+36\left(x+12\right)$$

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

$$\left(x+12\right)\left(x+36\right)$$

To find equation solutions, solve x+12=0 and x+36=0.

$$x=-12 x=-36$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -9 since division by zero is not defined. Multiply both sides of the equation by 2\left(x+9\right), the least common multiple of 2,2x+18.

$$\left(x+9\right)\left(x+18\right)+\left(x+9\right)x+2\left(x+9\right)\times 15=x^{2}+18x$$

Use the distributive property to multiply x+9 by x+18 and combine like terms.

$$x^{2}+27x+162+\left(x+9\right)x+2\left(x+9\right)\times 15=x^{2}+18x$$

Use the distributive property to multiply x+9 by x.

$$x^{2}+27x+162+x^{2}+9x+2\left(x+9\right)\times 15=x^{2}+18x$$

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

$$2x^{2}+27x+162+9x+2\left(x+9\right)\times 15=x^{2}+18x$$

Combine 27x and 9x to get 36x.

$$2x^{2}+36x+162+2\left(x+9\right)\times 15=x^{2}+18x$$

Multiply 2 and 15 to get 30.

$$2x^{2}+36x+162+30\left(x+9\right)=x^{2}+18x$$

Use the distributive property to multiply 30 by x+9.

$$2x^{2}+36x+162+30x+270=x^{2}+18x$$

Combine 36x and 30x to get 66x.

$$2x^{2}+66x+162+270=x^{2}+18x$$

Add 162 and 270 to get 432.

$$2x^{2}+66x+432=x^{2}+18x$$

Subtract x^{2} from both sides.

$$2x^{2}+66x+432-x^{2}=18x$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$x^{2}+66x+432=18x$$

Subtract 18x from both sides.

$$x^{2}+66x+432-18x=0$$

Combine 66x and -18x to get 48x.

$$x^{2}+48x+432=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 48 for b, and 432 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-48±\sqrt{48^{2}-4\times 432}}{2}$$

Square 48.

$$x=\frac{-48±\sqrt{2304-4\times 432}}{2}$$

Multiply -4 times 432.

$$x=\frac{-48±\sqrt{2304-1728}}{2}$$

Add 2304 to -1728.

$$x=\frac{-48±\sqrt{576}}{2}$$

Take the square root of 576.

$$x=\frac{-48±24}{2}$$

Now solve the equation x=\frac{-48±24}{2} when ± is plus. Add -48 to 24.

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

Divide -24 by 2.

$$x=-12$$

Now solve the equation x=\frac{-48±24}{2} when ± is minus. Subtract 24 from -48.

$$x=-\frac{72}{2}$$

Divide -72 by 2.

$$x=-36$$

The equation is now solved.

$$x=-12 x=-36$$

Steps for Completing the Square

Variable x cannot be equal to -9 since division by zero is not defined. Multiply both sides of the equation by 2\left(x+9\right), the least common multiple of 2,2x+18.

$$\left(x+9\right)\left(x+18\right)+\left(x+9\right)x+2\left(x+9\right)\times 15=x^{2}+18x$$

Use the distributive property to multiply x+9 by x+18 and combine like terms.

$$x^{2}+27x+162+\left(x+9\right)x+2\left(x+9\right)\times 15=x^{2}+18x$$

Use the distributive property to multiply x+9 by x.

$$x^{2}+27x+162+x^{2}+9x+2\left(x+9\right)\times 15=x^{2}+18x$$

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

$$2x^{2}+27x+162+9x+2\left(x+9\right)\times 15=x^{2}+18x$$

Combine 27x and 9x to get 36x.

$$2x^{2}+36x+162+2\left(x+9\right)\times 15=x^{2}+18x$$

Multiply 2 and 15 to get 30.

$$2x^{2}+36x+162+30\left(x+9\right)=x^{2}+18x$$

Use the distributive property to multiply 30 by x+9.

$$2x^{2}+36x+162+30x+270=x^{2}+18x$$

Combine 36x and 30x to get 66x.

$$2x^{2}+66x+162+270=x^{2}+18x$$

Add 162 and 270 to get 432.

$$2x^{2}+66x+432=x^{2}+18x$$

Subtract x^{2} from both sides.

$$2x^{2}+66x+432-x^{2}=18x$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$x^{2}+66x+432=18x$$

Subtract 18x from both sides.

$$x^{2}+66x+432-18x=0$$

Combine 66x and -18x to get 48x.

$$x^{2}+48x+432=0$$

Subtract 432 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}+48x=-432$$

Divide 48, the coefficient of the x term, by 2 to get 24. Then add the square of 24 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+48x+24^{2}=-432+24^{2}$$

Square 24.

$$x^{2}+48x+576=-432+576$$

Add -432 to 576.

$$x^{2}+48x+576=144$$

Factor x^{2}+48x+576. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+24\right)^{2}=144$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+24\right)^{2}}=\sqrt{144}$$

Simplify.

$$x+24=12 x+24=-12$$

Subtract 24 from both sides of the equation.

$$x=-12 x=-36$$