Solve 9x^2+12x-2

$$9 { x }^{ 2 } +12x-2$$

Factor

$$9\left(x-\frac{-\sqrt{6}-2}{3}\right)\left(x-\frac{\sqrt{6}-2}{3}\right)$$

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.

$$9x^{2}+12x-2=0$$

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.

$$x=\frac{-12±\sqrt{12^{2}-4\times 9\left(-2\right)}}{2\times 9}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 9\left(-2\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-12±\sqrt{144-36\left(-2\right)}}{2\times 9}$$

Multiply -36 times -2.

$$x=\frac{-12±\sqrt{144+72}}{2\times 9}$$

Add 144 to 72.

$$x=\frac{-12±\sqrt{216}}{2\times 9}$$

Take the square root of 216.

$$x=\frac{-12±6\sqrt{6}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{-12±6\sqrt{6}}{18}$$

Now solve the equation x=\frac{-12±6\sqrt{6}}{18} when ± is plus. Add -12 to 6\sqrt{6}.

$$x=\frac{6\sqrt{6}-12}{18}$$

Divide -12+6\sqrt{6} by 18.

$$x=\frac{\sqrt{6}-2}{3}$$

Now solve the equation x=\frac{-12±6\sqrt{6}}{18} when ± is minus. Subtract 6\sqrt{6} from -12.

$$x=\frac{-6\sqrt{6}-12}{18}$$

Divide -12-6\sqrt{6} by 18.

$$x=\frac{-\sqrt{6}-2}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-2+\sqrt{6}}{3} for x_{1} and \frac{-2-\sqrt{6}}{3} for x_{2}.

$$9x^{2}+12x-2=9\left(x-\frac{\sqrt{6}-2}{3}\right)\left(x-\frac{-\sqrt{6}-2}{3}\right)$$

Evaluate

$$9x^{2}+12x-2$$