Solve 6x^2+6x-18

$$6 { x }^{ 2 } +6x-18$$

Factor

$$6\left(x-\frac{-\sqrt{13}-1}{2}\right)\left(x-\frac{\sqrt{13}-1}{2}\right)$$

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.

$$6x^{2}+6x-18=0$$

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.

$$x=\frac{-6±\sqrt{6^{2}-4\times 6\left(-18\right)}}{2\times 6}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\times 6\left(-18\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-6±\sqrt{36-24\left(-18\right)}}{2\times 6}$$

Multiply -24 times -18.

$$x=\frac{-6±\sqrt{36+432}}{2\times 6}$$

Add 36 to 432.

$$x=\frac{-6±\sqrt{468}}{2\times 6}$$

Take the square root of 468.

$$x=\frac{-6±6\sqrt{13}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-6±6\sqrt{13}}{12}$$

Now solve the equation x=\frac{-6±6\sqrt{13}}{12} when ± is plus. Add -6 to 6\sqrt{13}.

$$x=\frac{6\sqrt{13}-6}{12}$$

Divide -6+6\sqrt{13} by 12.

$$x=\frac{\sqrt{13}-1}{2}$$

Now solve the equation x=\frac{-6±6\sqrt{13}}{12} when ± is minus. Subtract 6\sqrt{13} from -6.

$$x=\frac{-6\sqrt{13}-6}{12}$$

Divide -6-6\sqrt{13} by 12.

$$x=\frac{-\sqrt{13}-1}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-1+\sqrt{13}}{2} for x_{1} and \frac{-1-\sqrt{13}}{2} for x_{2}.

$$6x^{2}+6x-18=6\left(x-\frac{\sqrt{13}-1}{2}\right)\left(x-\frac{-\sqrt{13}-1}{2}\right)$$

Evaluate

$$6\left(x^{2}+x-3\right)$$