Solve 6x^2-3x-36

$$6 { x }^{ 2 } -3x-36$$

Factor

$$6\left(x-\frac{1-\sqrt{97}}{4}\right)\left(x-\frac{\sqrt{97}+1}{4}\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$6x^{2}-3x-36=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 6\left(-36\right)}}{2\times 6}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 6\left(-36\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-3\right)±\sqrt{9-24\left(-36\right)}}{2\times 6}$$

Multiply -24 times -36.

$$x=\frac{-\left(-3\right)±\sqrt{9+864}}{2\times 6}$$

Add 9 to 864.

$$x=\frac{-\left(-3\right)±\sqrt{873}}{2\times 6}$$

Take the square root of 873.

$$x=\frac{-\left(-3\right)±3\sqrt{97}}{2\times 6}$$

The opposite of -3 is 3.

$$x=\frac{3±3\sqrt{97}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{3±3\sqrt{97}}{12}$$

Now solve the equation x=\frac{3±3\sqrt{97}}{12} when ± is plus. Add 3 to 3\sqrt{97}.

$$x=\frac{3\sqrt{97}+3}{12}$$

Divide 3+3\sqrt{97} by 12.

$$x=\frac{\sqrt{97}+1}{4}$$

Now solve the equation x=\frac{3±3\sqrt{97}}{12} when ± is minus. Subtract 3\sqrt{97} from 3.

$$x=\frac{3-3\sqrt{97}}{12}$$

Divide 3-3\sqrt{97} by 12.

$$x=\frac{1-\sqrt{97}}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1+\sqrt{97}}{4} for x_{1} and \frac{1-\sqrt{97}}{4} for x_{2}.

$$6x^{2}-3x-36=6\left(x-\frac{\sqrt{97}+1}{4}\right)\left(x-\frac{1-\sqrt{97}}{4}\right)$$

Evaluate

$$6x^{2}-3x-36$$