Solve 6x^2-x-22x=0

$$6 x ^ { 2 } - x - 22 x = 0$$

Solve for x

$$x = \frac{23}{6} = 3\frac{5}{6} \approx 3.833333333$$

$$x=0$$

Steps Using Factoring

Factor out x.

$$x\left(6x-1-22\right)=0$$

To find equation solutions, solve x=0 and 6x-23=0.

$$x=0 x=\frac{23}{6}$$

Steps Using the Quadratic Formula

Combine -x and -22x to get -23x.

$$6x^{2}-23x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -23 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-23\right)±\sqrt{\left(-23\right)^{2}}}{2\times 6}$$

Take the square root of \left(-23\right)^{2}.

$$x=\frac{-\left(-23\right)±23}{2\times 6}$$

The opposite of -23 is 23.

$$x=\frac{23±23}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{23±23}{12}$$

Now solve the equation x=\frac{23±23}{12} when ± is plus. Add 23 to 23.

$$x=\frac{46}{12}$$

Reduce the fraction \frac{46}{12} to lowest terms by extracting and canceling out 2.

$$x=\frac{23}{6}$$

Now solve the equation x=\frac{23±23}{12} when ± is minus. Subtract 23 from 23.

$$x=\frac{0}{12}$$

Divide 0 by 12.

$$x=0$$

The equation is now solved.

$$x=\frac{23}{6} x=0$$

Steps for Completing the Square

Combine -x and -22x to get -23x.

$$6x^{2}-23x=0$$

Divide both sides by 6.

$$\frac{6x^{2}-23x}{6}=\frac{0}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}-\frac{23}{6}x=\frac{0}{6}$$

Divide 0 by 6.

$$x^{2}-\frac{23}{6}x=0$$

Divide -\frac{23}{6}, the coefficient of the x term, by 2 to get -\frac{23}{12}. Then add the square of -\frac{23}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{23}{6}x+\left(-\frac{23}{12}\right)^{2}=\left(-\frac{23}{12}\right)^{2}$$

Square -\frac{23}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{23}{6}x+\frac{529}{144}=\frac{529}{144}$$

Factor x^{2}-\frac{23}{6}x+\frac{529}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{23}{12}\right)^{2}=\frac{529}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{23}{12}\right)^{2}}=\sqrt{\frac{529}{144}}$$

Simplify.

$$x-\frac{23}{12}=\frac{23}{12} x-\frac{23}{12}=-\frac{23}{12}$$

Add \frac{23}{12} to both sides of the equation.

$$x=\frac{23}{6} x=0$$