Solve 0=x^2+31.4x-376.75

$$0 = { x }^{ 2 } +31.4x-376.75$$

Solve for x

$$x=\frac{\sqrt{15581}}{5}-15.7\approx 9.264775184$$

$$x=-\frac{\sqrt{15581}}{5}-15.7\approx -40.664775184$$

Steps Using the Quadratic Formula

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

$$x^{2}+31.4x-376.75=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 31.4 for b, and -376.75 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-31.4±\sqrt{31.4^{2}-4\left(-376.75\right)}}{2}$$

Square 31.4 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-31.4±\sqrt{985.96-4\left(-376.75\right)}}{2}$$

Multiply -4 times -376.75.

$$x=\frac{-31.4±\sqrt{985.96+1507}}{2}$$

Add 985.96 to 1507.

$$x=\frac{-31.4±\sqrt{2492.96}}{2}$$

Take the square root of 2492.96.

$$x=\frac{-31.4±\frac{2\sqrt{15581}}{5}}{2}$$

Now solve the equation x=\frac{-31.4±\frac{2\sqrt{15581}}{5}}{2} when ± is plus. Add -31.4 to \frac{2\sqrt{15581}}{5}.

$$x=\frac{2\sqrt{15581}-157}{2\times 5}$$

Divide \frac{-157+2\sqrt{15581}}{5} by 2.

$$x=\frac{\sqrt{15581}}{5}-\frac{157}{10}$$

Now solve the equation x=\frac{-31.4±\frac{2\sqrt{15581}}{5}}{2} when ± is minus. Subtract \frac{2\sqrt{15581}}{5} from -31.4.

$$x=\frac{-2\sqrt{15581}-157}{2\times 5}$$

Divide \frac{-157-2\sqrt{15581}}{5} by 2.

$$x=-\frac{\sqrt{15581}}{5}-\frac{157}{10}$$

The equation is now solved.

$$x=\frac{\sqrt{15581}}{5}-\frac{157}{10} x=-\frac{\sqrt{15581}}{5}-\frac{157}{10}$$

Steps for Completing the Square

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

$$x^{2}+31.4x-376.75=0$$

Add 376.75 to both sides. Anything plus zero gives itself.

$$x^{2}+31.4x=376.75$$

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.

$$x^{2}+31.4x=\frac{1507}{4}$$

Divide 31.4, the coefficient of the x term, by 2 to get 15.7. Then add the square of 15.7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+31.4x+15.7^{2}=\frac{1507}{4}+15.7^{2}$$

Square 15.7 by squaring both the numerator and the denominator of the fraction.

$$x^{2}+31.4x+246.49=\frac{1507}{4}+246.49$$

Add \frac{1507}{4} to 246.49 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+31.4x+246.49=\frac{15581}{25}$$

Factor x^{2}+31.4x+246.49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+15.7\right)^{2}=\frac{15581}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+15.7\right)^{2}}=\sqrt{\frac{15581}{25}}$$

Simplify.

$$x+15.7=\frac{\sqrt{15581}}{5} x+15.7=-\frac{\sqrt{15581}}{5}$$

Subtract 15.7 from both sides of the equation.

$$x=\frac{\sqrt{15581}}{5}-\frac{157}{10} x=-\frac{\sqrt{15581}}{5}-\frac{157}{10}$$