Solve (x-5)x=208

$$(x-5)x=208$$

Solve for x

$$x = \frac{\sqrt{857} + 5}{2} \approx 17.137281168$$

$$x=\frac{5-\sqrt{857}}{2}\approx -12.137281168$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-5 by x.

$$x^{2}-5x=208$$

Subtract 208 from both sides.

$$x^{2}-5x-208=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -5 for b, and -208 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-208\right)}}{2}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\left(-208\right)}}{2}$$

Multiply -4 times -208.

$$x=\frac{-\left(-5\right)±\sqrt{25+832}}{2}$$

Add 25 to 832.

$$x=\frac{-\left(-5\right)±\sqrt{857}}{2}$$

The opposite of -5 is 5.

$$x=\frac{5±\sqrt{857}}{2}$$

Now solve the equation x=\frac{5±\sqrt{857}}{2} when ± is plus. Add 5 to \sqrt{857}.

$$x=\frac{\sqrt{857}+5}{2}$$

Now solve the equation x=\frac{5±\sqrt{857}}{2} when ± is minus. Subtract \sqrt{857} from 5.

$$x=\frac{5-\sqrt{857}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{857}+5}{2} x=\frac{5-\sqrt{857}}{2}$$

Steps for Completing the Square

Use the distributive property to multiply x-5 by x.

$$x^{2}-5x=208$$

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

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=208+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-5x+\frac{25}{4}=208+\frac{25}{4}$$

Add 208 to \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=\frac{857}{4}$$

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

$$\left(x-\frac{5}{2}\right)^{2}=\frac{857}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{857}{4}}$$

Simplify.

$$x-\frac{5}{2}=\frac{\sqrt{857}}{2} x-\frac{5}{2}=-\frac{\sqrt{857}}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$x=\frac{\sqrt{857}+5}{2} x=\frac{5-\sqrt{857}}{2}$$