Solve 6x^2=x+28

$$6 x ^ { 2 } = x + 28$$

Solve for x

$$x = \frac{\sqrt{673} + 1}{12} \approx 2.245186962$$

$$x=\frac{1-\sqrt{673}}{12}\approx -2.078520295$$

Steps Using the Quadratic Formula

Subtract x from both sides.

$$6x^{2}-x=28$$

Subtract 28 from both sides.

$$6x^{2}-x-28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -1 for b, and -28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 6\left(-28\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-1\right)±\sqrt{1-24\left(-28\right)}}{2\times 6}$$

Multiply -24 times -28.

$$x=\frac{-\left(-1\right)±\sqrt{1+672}}{2\times 6}$$

Add 1 to 672.

$$x=\frac{-\left(-1\right)±\sqrt{673}}{2\times 6}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{673}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{1±\sqrt{673}}{12}$$

Now solve the equation x=\frac{1±\sqrt{673}}{12} when ± is plus. Add 1 to \sqrt{673}.

$$x=\frac{\sqrt{673}+1}{12}$$

Now solve the equation x=\frac{1±\sqrt{673}}{12} when ± is minus. Subtract \sqrt{673} from 1.

$$x=\frac{1-\sqrt{673}}{12}$$

The equation is now solved.

$$x=\frac{\sqrt{673}+1}{12} x=\frac{1-\sqrt{673}}{12}$$

Steps for Completing the Square

Subtract x from both sides.

$$6x^{2}-x=28$$

Divide both sides by 6.

$$\frac{6x^{2}-x}{6}=\frac{28}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}-\frac{1}{6}x=\frac{28}{6}$$

Reduce the fraction \frac{28}{6} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{1}{6}x=\frac{14}{3}$$

Divide -\frac{1}{6}, the coefficient of the x term, by 2 to get -\frac{1}{12}. Then add the square of -\frac{1}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{6}x+\left(-\frac{1}{12}\right)^{2}=\frac{14}{3}+\left(-\frac{1}{12}\right)^{2}$$

Square -\frac{1}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{6}x+\frac{1}{144}=\frac{14}{3}+\frac{1}{144}$$

Add \frac{14}{3} to \frac{1}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{6}x+\frac{1}{144}=\frac{673}{144}$$

Factor x^{2}-\frac{1}{6}x+\frac{1}{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}.

$$\left(x-\frac{1}{12}\right)^{2}=\frac{673}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{12}\right)^{2}}=\sqrt{\frac{673}{144}}$$

Simplify.

$$x-\frac{1}{12}=\frac{\sqrt{673}}{12} x-\frac{1}{12}=-\frac{\sqrt{673}}{12}$$

Add \frac{1}{12} to both sides of the equation.

$$x=\frac{\sqrt{673}+1}{12} x=\frac{1-\sqrt{673}}{12}$$