Solve 5x^2=111x-2

$$5 x ^ { 2 } = 111 x - 2$$

Solve for x

$$x = \frac{\sqrt{12281} + 111}{10} \approx 22.181967334$$

$$x=\frac{111-\sqrt{12281}}{10}\approx 0.018032666$$

Steps Using the Quadratic Formula

Subtract 111x from both sides.

$$5x^{2}-111x=-2$$

Add 2 to both sides.

$$5x^{2}-111x+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -111 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-111\right)±\sqrt{\left(-111\right)^{2}-4\times 5\times 2}}{2\times 5}$$

Square -111.

$$x=\frac{-\left(-111\right)±\sqrt{12321-4\times 5\times 2}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-111\right)±\sqrt{12321-20\times 2}}{2\times 5}$$

Multiply -20 times 2.

$$x=\frac{-\left(-111\right)±\sqrt{12321-40}}{2\times 5}$$

Add 12321 to -40.

$$x=\frac{-\left(-111\right)±\sqrt{12281}}{2\times 5}$$

The opposite of -111 is 111.

$$x=\frac{111±\sqrt{12281}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{111±\sqrt{12281}}{10}$$

Now solve the equation x=\frac{111±\sqrt{12281}}{10} when ± is plus. Add 111 to \sqrt{12281}.

$$x=\frac{\sqrt{12281}+111}{10}$$

Now solve the equation x=\frac{111±\sqrt{12281}}{10} when ± is minus. Subtract \sqrt{12281} from 111.

$$x=\frac{111-\sqrt{12281}}{10}$$

The equation is now solved.

$$x=\frac{\sqrt{12281}+111}{10} x=\frac{111-\sqrt{12281}}{10}$$

Steps for Completing the Square

Subtract 111x from both sides.

$$5x^{2}-111x=-2$$

Divide both sides by 5.

$$\frac{5x^{2}-111x}{5}=-\frac{2}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}-\frac{111}{5}x=-\frac{2}{5}$$

Divide -\frac{111}{5}, the coefficient of the x term, by 2 to get -\frac{111}{10}. Then add the square of -\frac{111}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{111}{5}x+\left(-\frac{111}{10}\right)^{2}=-\frac{2}{5}+\left(-\frac{111}{10}\right)^{2}$$

Square -\frac{111}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{111}{5}x+\frac{12321}{100}=-\frac{2}{5}+\frac{12321}{100}$$

Add -\frac{2}{5} to \frac{12321}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{111}{5}x+\frac{12321}{100}=\frac{12281}{100}$$

Factor x^{2}-\frac{111}{5}x+\frac{12321}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{111}{10}\right)^{2}=\frac{12281}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{111}{10}\right)^{2}}=\sqrt{\frac{12281}{100}}$$

Simplify.

$$x-\frac{111}{10}=\frac{\sqrt{12281}}{10} x-\frac{111}{10}=-\frac{\sqrt{12281}}{10}$$

Add \frac{111}{10} to both sides of the equation.

$$x=\frac{\sqrt{12281}+111}{10} x=\frac{111-\sqrt{12281}}{10}$$