Solve 636=x/2(18+(x-1)8)

$$636= \frac{ x }{ 2 } (18+(x-1)8)$$

Solve for x

$$x = -\frac{53}{4} = -13\frac{1}{4} = -13.25$$

$$x=12$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$1272=x\left(18+\left(x-1\right)\times 8\right)$$

Use the distributive property to multiply x-1 by 8.

$$1272=x\left(18+8x-8\right)$$

Subtract 8 from 18 to get 10.

$$1272=x\left(10+8x\right)$$

Use the distributive property to multiply x by 10+8x.

$$1272=10x+8x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$10x+8x^{2}=1272$$

Subtract 1272 from both sides.

$$10x+8x^{2}-1272=0$$

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.

$$8x^{2}+10x-1272=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, 10 for b, and -1272 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\times 8\left(-1272\right)}}{2\times 8}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\times 8\left(-1272\right)}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-10±\sqrt{100-32\left(-1272\right)}}{2\times 8}$$

Multiply -32 times -1272.

$$x=\frac{-10±\sqrt{100+40704}}{2\times 8}$$

Add 100 to 40704.

$$x=\frac{-10±\sqrt{40804}}{2\times 8}$$

Take the square root of 40804.

$$x=\frac{-10±202}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{-10±202}{16}$$

Now solve the equation x=\frac{-10±202}{16} when ± is plus. Add -10 to 202.

$$x=\frac{192}{16}$$

Divide 192 by 16.

$$x=12$$

Now solve the equation x=\frac{-10±202}{16} when ± is minus. Subtract 202 from -10.

$$x=-\frac{212}{16}$$

Reduce the fraction \frac{-212}{16} to lowest terms by extracting and canceling out 4.

$$x=-\frac{53}{4}$$

The equation is now solved.

$$x=12 x=-\frac{53}{4}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$1272=x\left(18+\left(x-1\right)\times 8\right)$$

Use the distributive property to multiply x-1 by 8.

$$1272=x\left(18+8x-8\right)$$

Subtract 8 from 18 to get 10.

$$1272=x\left(10+8x\right)$$

Use the distributive property to multiply x by 10+8x.

$$1272=10x+8x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$10x+8x^{2}=1272$$

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.

$$8x^{2}+10x=1272$$

Divide both sides by 8.

$$\frac{8x^{2}+10x}{8}=\frac{1272}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\frac{10}{8}x=\frac{1272}{8}$$

Reduce the fraction \frac{10}{8} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{5}{4}x=\frac{1272}{8}$$

Divide 1272 by 8.

$$x^{2}+\frac{5}{4}x=159$$

Divide \frac{5}{4}, the coefficient of the x term, by 2 to get \frac{5}{8}. Then add the square of \frac{5}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{5}{4}x+\left(\frac{5}{8}\right)^{2}=159+\left(\frac{5}{8}\right)^{2}$$

Square \frac{5}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5}{4}x+\frac{25}{64}=159+\frac{25}{64}$$

Add 159 to \frac{25}{64}.

$$x^{2}+\frac{5}{4}x+\frac{25}{64}=\frac{10201}{64}$$

Factor x^{2}+\frac{5}{4}x+\frac{25}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{8}\right)^{2}=\frac{10201}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{8}\right)^{2}}=\sqrt{\frac{10201}{64}}$$

Simplify.

$$x+\frac{5}{8}=\frac{101}{8} x+\frac{5}{8}=-\frac{101}{8}$$

Subtract \frac{5}{8} from both sides of the equation.

$$x=12 x=-\frac{53}{4}$$