Solve 0=20x^2-187x+222

$$0 = 20 x ^ { 2 } - 187 x + 222$$

Solve for x

$$x = \frac{\sqrt{17209} + 187}{40} \approx 7.954576954$$

$$x = \frac{187 - \sqrt{17209}}{40} \approx 1.395423046$$

Steps Using the Quadratic Formula

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

$$20x^{2}-187x+222=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 20 for a, -187 for b, and 222 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-187\right)±\sqrt{\left(-187\right)^{2}-4\times 20\times 222}}{2\times 20}$$

Square -187.

$$x=\frac{-\left(-187\right)±\sqrt{34969-4\times 20\times 222}}{2\times 20}$$

Multiply -4 times 20.

$$x=\frac{-\left(-187\right)±\sqrt{34969-80\times 222}}{2\times 20}$$

Multiply -80 times 222.

$$x=\frac{-\left(-187\right)±\sqrt{34969-17760}}{2\times 20}$$

Add 34969 to -17760.

$$x=\frac{-\left(-187\right)±\sqrt{17209}}{2\times 20}$$

The opposite of -187 is 187.

$$x=\frac{187±\sqrt{17209}}{2\times 20}$$

Multiply 2 times 20.

$$x=\frac{187±\sqrt{17209}}{40}$$

Now solve the equation x=\frac{187±\sqrt{17209}}{40} when ± is plus. Add 187 to \sqrt{17209}.

$$x=\frac{\sqrt{17209}+187}{40}$$

Now solve the equation x=\frac{187±\sqrt{17209}}{40} when ± is minus. Subtract \sqrt{17209} from 187.

$$x=\frac{187-\sqrt{17209}}{40}$$

The equation is now solved.

$$x=\frac{\sqrt{17209}+187}{40} x=\frac{187-\sqrt{17209}}{40}$$

Steps for Completing the Square

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

$$20x^{2}-187x+222=0$$

Subtract 222 from both sides. Anything subtracted from zero gives its negation.

$$20x^{2}-187x=-222$$

Divide both sides by 20.

$$\frac{20x^{2}-187x}{20}=-\frac{222}{20}$$

Dividing by 20 undoes the multiplication by 20.

$$x^{2}-\frac{187}{20}x=-\frac{222}{20}$$

Reduce the fraction \frac{-222}{20} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{187}{20}x=-\frac{111}{10}$$

Divide -\frac{187}{20}, the coefficient of the x term, by 2 to get -\frac{187}{40}. Then add the square of -\frac{187}{40} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{187}{20}x+\left(-\frac{187}{40}\right)^{2}=-\frac{111}{10}+\left(-\frac{187}{40}\right)^{2}$$

Square -\frac{187}{40} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{187}{20}x+\frac{34969}{1600}=-\frac{111}{10}+\frac{34969}{1600}$$

Add -\frac{111}{10} to \frac{34969}{1600} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{187}{20}x+\frac{34969}{1600}=\frac{17209}{1600}$$

Factor x^{2}-\frac{187}{20}x+\frac{34969}{1600}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{187}{40}\right)^{2}=\frac{17209}{1600}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{187}{40}\right)^{2}}=\sqrt{\frac{17209}{1600}}$$

Simplify.

$$x-\frac{187}{40}=\frac{\sqrt{17209}}{40} x-\frac{187}{40}=-\frac{\sqrt{17209}}{40}$$

Add \frac{187}{40} to both sides of the equation.

$$x=\frac{\sqrt{17209}+187}{40} x=\frac{187-\sqrt{17209}}{40}$$