Solve 108=xleft(x+10right)

$$108=x \left( x+10 \right)$$

Solve for x (complex solution)

$$x=\sqrt{133}-5\approx 6.532562595$$

$$x=-\left(\sqrt{133}+5\right)\approx -16.532562595$$

Steps Using the Quadratic Formula

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

$$108=x^{2}+10x$$

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

$$x^{2}+10x=108$$

Subtract 108 from both sides.

$$x^{2}+10x-108=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 10 for b, and -108 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-108\right)}}{2}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\left(-108\right)}}{2}$$

Multiply -4 times -108.

$$x=\frac{-10±\sqrt{100+432}}{2}$$

Add 100 to 432.

$$x=\frac{-10±\sqrt{532}}{2}$$

Take the square root of 532.

$$x=\frac{-10±2\sqrt{133}}{2}$$

Now solve the equation x=\frac{-10±2\sqrt{133}}{2} when ± is plus. Add -10 to 2\sqrt{133}.

$$x=\frac{2\sqrt{133}-10}{2}$$

Divide -10+2\sqrt{133} by 2.

$$x=\sqrt{133}-5$$

Now solve the equation x=\frac{-10±2\sqrt{133}}{2} when ± is minus. Subtract 2\sqrt{133} from -10.

$$x=\frac{-2\sqrt{133}-10}{2}$$

Divide -10-2\sqrt{133} by 2.

$$x=-\sqrt{133}-5$$

The equation is now solved.

$$x=\sqrt{133}-5 x=-\sqrt{133}-5$$

Steps for Completing the Square

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

$$108=x^{2}+10x$$

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

$$x^{2}+10x=108$$

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.

$$x^{2}+10x+5^{2}=108+5^{2}$$

Square 5.

$$x^{2}+10x+25=108+25$$

Add 108 to 25.

$$x^{2}+10x+25=133$$

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}.

$$\left(x+5\right)^{2}=133$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{133}$$

Simplify.

$$x+5=\sqrt{133} x+5=-\sqrt{133}$$

Subtract 5 from both sides of the equation.

$$x=\sqrt{133}-5 x=-\sqrt{133}-5$$

Solve for x

$$x=\sqrt{133}-5\approx 6.532562595$$

$$x=-\sqrt{133}-5\approx -16.532562595$$

Steps Using the Quadratic Formula

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

$$108=x^{2}+10x$$

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

$$x^{2}+10x=108$$

Subtract 108 from both sides.

$$x^{2}+10x-108=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 10 for b, and -108 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-108\right)}}{2}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\left(-108\right)}}{2}$$

Multiply -4 times -108.

$$x=\frac{-10±\sqrt{100+432}}{2}$$

Add 100 to 432.

$$x=\frac{-10±\sqrt{532}}{2}$$

Take the square root of 532.

$$x=\frac{-10±2\sqrt{133}}{2}$$

Now solve the equation x=\frac{-10±2\sqrt{133}}{2} when ± is plus. Add -10 to 2\sqrt{133}.

$$x=\frac{2\sqrt{133}-10}{2}$$

Divide -10+2\sqrt{133} by 2.

$$x=\sqrt{133}-5$$

Now solve the equation x=\frac{-10±2\sqrt{133}}{2} when ± is minus. Subtract 2\sqrt{133} from -10.

$$x=\frac{-2\sqrt{133}-10}{2}$$

Divide -10-2\sqrt{133} by 2.

$$x=-\sqrt{133}-5$$

The equation is now solved.

$$x=\sqrt{133}-5 x=-\sqrt{133}-5$$

Steps for Completing the Square

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

$$108=x^{2}+10x$$

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

$$x^{2}+10x=108$$

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.

$$x^{2}+10x+5^{2}=108+5^{2}$$

Square 5.

$$x^{2}+10x+25=108+25$$

Add 108 to 25.

$$x^{2}+10x+25=133$$

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}.

$$\left(x+5\right)^{2}=133$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{133}$$

Simplify.

$$x+5=\sqrt{133} x+5=-\sqrt{133}$$

Subtract 5 from both sides of the equation.

$$x=\sqrt{133}-5 x=-\sqrt{133}-5$$