Solve 125x^2+x-12=19x

$$125 x ^ { 2 } + x - 12 = 19 x$$

Solve for x

$$x=\frac{\sqrt{1581}+9}{125}\approx 0.390094326$$

$$x=\frac{9-\sqrt{1581}}{125}\approx -0.246094326$$

Steps Using the Quadratic Formula

Subtract 19x from both sides.

$$125x^{2}+x-12-19x=0$$

Combine x and -19x to get -18x.

$$125x^{2}-18x-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 125 for a, -18 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 125\left(-12\right)}}{2\times 125}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 125\left(-12\right)}}{2\times 125}$$

Multiply -4 times 125.

$$x=\frac{-\left(-18\right)±\sqrt{324-500\left(-12\right)}}{2\times 125}$$

Multiply -500 times -12.

$$x=\frac{-\left(-18\right)±\sqrt{324+6000}}{2\times 125}$$

Add 324 to 6000.

$$x=\frac{-\left(-18\right)±\sqrt{6324}}{2\times 125}$$

Take the square root of 6324.

$$x=\frac{-\left(-18\right)±2\sqrt{1581}}{2\times 125}$$

The opposite of -18 is 18.

$$x=\frac{18±2\sqrt{1581}}{2\times 125}$$

Multiply 2 times 125.

$$x=\frac{18±2\sqrt{1581}}{250}$$

Now solve the equation x=\frac{18±2\sqrt{1581}}{250} when ± is plus. Add 18 to 2\sqrt{1581}.

$$x=\frac{2\sqrt{1581}+18}{250}$$

Divide 18+2\sqrt{1581} by 250.

$$x=\frac{\sqrt{1581}+9}{125}$$

Now solve the equation x=\frac{18±2\sqrt{1581}}{250} when ± is minus. Subtract 2\sqrt{1581} from 18.

$$x=\frac{18-2\sqrt{1581}}{250}$$

Divide 18-2\sqrt{1581} by 250.

$$x=\frac{9-\sqrt{1581}}{125}$$

The equation is now solved.

$$x=\frac{\sqrt{1581}+9}{125} x=\frac{9-\sqrt{1581}}{125}$$

Steps for Completing the Square

Subtract 19x from both sides.

$$125x^{2}+x-12-19x=0$$

Combine x and -19x to get -18x.

$$125x^{2}-18x-12=0$$

Add 12 to both sides. Anything plus zero gives itself.

$$125x^{2}-18x=12$$

Divide both sides by 125.

$$\frac{125x^{2}-18x}{125}=\frac{12}{125}$$

Dividing by 125 undoes the multiplication by 125.

$$x^{2}-\frac{18}{125}x=\frac{12}{125}$$

Divide -\frac{18}{125}, the coefficient of the x term, by 2 to get -\frac{9}{125}. Then add the square of -\frac{9}{125} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{18}{125}x+\left(-\frac{9}{125}\right)^{2}=\frac{12}{125}+\left(-\frac{9}{125}\right)^{2}$$

Square -\frac{9}{125} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{18}{125}x+\frac{81}{15625}=\frac{12}{125}+\frac{81}{15625}$$

Add \frac{12}{125} to \frac{81}{15625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{18}{125}x+\frac{81}{15625}=\frac{1581}{15625}$$

Factor x^{2}-\frac{18}{125}x+\frac{81}{15625}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{125}\right)^{2}=\frac{1581}{15625}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{125}\right)^{2}}=\sqrt{\frac{1581}{15625}}$$

Simplify.

$$x-\frac{9}{125}=\frac{\sqrt{1581}}{125} x-\frac{9}{125}=-\frac{\sqrt{1581}}{125}$$

Add \frac{9}{125} to both sides of the equation.

$$x=\frac{\sqrt{1581}+9}{125} x=\frac{9-\sqrt{1581}}{125}$$