Solve 56=-32.139x^2+13.089x+71.856

$$56 = -32.139 { x }^{ 2 } +13.089x+71.856$$

Solve for x

$$x=\frac{\sqrt{245522873}+4363}{21426}\approx 0.934946771$$

$$x=\frac{4363-\sqrt{245522873}}{21426}\approx -0.527684566$$

Steps Using the Quadratic Formula

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

$$-32.139x^{2}+13.089x+71.856=56$$

Subtract 56 from both sides.

$$-32.139x^{2}+13.089x+71.856-56=0$$

Subtract 56 from 71.856 to get 15.856.

$$-32.139x^{2}+13.089x+15.856=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -32.139 for a, 13.089 for b, and 15.856 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13.089±\sqrt{13.089^{2}-4\left(-32.139\right)\times 15.856}}{2\left(-32.139\right)}$$

Square 13.089 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-13.089±\sqrt{171.321921-4\left(-32.139\right)\times 15.856}}{2\left(-32.139\right)}$$

Multiply -4 times -32.139.

$$x=\frac{-13.089±\sqrt{171.321921+128.556\times 15.856}}{2\left(-32.139\right)}$$

Multiply 128.556 times 15.856 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-13.089±\sqrt{171.321921+2038.383936}}{2\left(-32.139\right)}$$

Add 171.321921 to 2038.383936 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-13.089±\sqrt{2209.705857}}{2\left(-32.139\right)}$$

Take the square root of 2209.705857.

$$x=\frac{-13.089±\frac{3\sqrt{245522873}}{1000}}{2\left(-32.139\right)}$$

Multiply 2 times -32.139.

$$x=\frac{-13.089±\frac{3\sqrt{245522873}}{1000}}{-64.278}$$

Now solve the equation x=\frac{-13.089±\frac{3\sqrt{245522873}}{1000}}{-64.278} when ± is plus. Add -13.089 to \frac{3\sqrt{245522873}}{1000}.

$$x=\frac{3\sqrt{245522873}-13089}{-64.278\times 1000}$$

Divide \frac{-13089+3\sqrt{245522873}}{1000} by -64.278 by multiplying \frac{-13089+3\sqrt{245522873}}{1000} by the reciprocal of -64.278.

$$x=\frac{4363-\sqrt{245522873}}{21426}$$

Now solve the equation x=\frac{-13.089±\frac{3\sqrt{245522873}}{1000}}{-64.278} when ± is minus. Subtract \frac{3\sqrt{245522873}}{1000} from -13.089.

$$x=\frac{-3\sqrt{245522873}-13089}{-64.278\times 1000}$$

Divide \frac{-13089-3\sqrt{245522873}}{1000} by -64.278 by multiplying \frac{-13089-3\sqrt{245522873}}{1000} by the reciprocal of -64.278.

$$x=\frac{\sqrt{245522873}+4363}{21426}$$

The equation is now solved.

$$x=\frac{4363-\sqrt{245522873}}{21426} x=\frac{\sqrt{245522873}+4363}{21426}$$

Steps for Completing the Square

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

$$-32.139x^{2}+13.089x+71.856=56$$

Subtract 71.856 from both sides.

$$-32.139x^{2}+13.089x=56-71.856$$

Subtract 71.856 from 56 to get -15.856.

$$-32.139x^{2}+13.089x=-15.856$$

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.

$$-32.139x^{2}+13.089x=-\frac{1982}{125}$$

Divide both sides of the equation by -32.139, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-32.139x^{2}+13.089x}{-32.139}=-\frac{\frac{1982}{125}}{-32.139}$$

Dividing by -32.139 undoes the multiplication by -32.139.

$$x^{2}+\frac{13.089}{-32.139}x=-\frac{\frac{1982}{125}}{-32.139}$$

Divide 13.089 by -32.139 by multiplying 13.089 by the reciprocal of -32.139.

$$x^{2}-\frac{4363}{10713}x=-\frac{\frac{1982}{125}}{-32.139}$$

Divide -\frac{1982}{125} by -32.139 by multiplying -\frac{1982}{125} by the reciprocal of -32.139.

$$x^{2}-\frac{4363}{10713}x=\frac{15856}{32139}$$

Divide -\frac{4363}{10713}, the coefficient of the x term, by 2 to get -\frac{4363}{21426}. Then add the square of -\frac{4363}{21426} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{4363}{10713}x+\left(-\frac{4363}{21426}\right)^{2}=\frac{15856}{32139}+\left(-\frac{4363}{21426}\right)^{2}$$

Square -\frac{4363}{21426} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{4363}{10713}x+\frac{19035769}{459073476}=\frac{15856}{32139}+\frac{19035769}{459073476}$$

Add \frac{15856}{32139} to \frac{19035769}{459073476} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{4363}{10713}x+\frac{19035769}{459073476}=\frac{245522873}{459073476}$$

Factor x^{2}-\frac{4363}{10713}x+\frac{19035769}{459073476}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{4363}{21426}\right)^{2}=\frac{245522873}{459073476}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4363}{21426}\right)^{2}}=\sqrt{\frac{245522873}{459073476}}$$

Simplify.

$$x-\frac{4363}{21426}=\frac{\sqrt{245522873}}{21426} x-\frac{4363}{21426}=-\frac{\sqrt{245522873}}{21426}$$

Add \frac{4363}{21426} to both sides of the equation.

$$x=\frac{\sqrt{245522873}+4363}{21426} x=\frac{4363-\sqrt{245522873}}{21426}$$