Solve 9x^2-6x-18

$$9 { x }^{ 2 } -6x-18$$

Factor

$$9\left(x-\frac{1-\sqrt{19}}{3}\right)\left(x-\frac{\sqrt{19}+1}{3}\right)$$

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.

$$9x^{2}-6x-18=0$$

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.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 9\left(-18\right)}}{2\times 9}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 9\left(-18\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-6\right)±\sqrt{36-36\left(-18\right)}}{2\times 9}$$

Multiply -36 times -18.

$$x=\frac{-\left(-6\right)±\sqrt{36+648}}{2\times 9}$$

Add 36 to 648.

$$x=\frac{-\left(-6\right)±\sqrt{684}}{2\times 9}$$

Take the square root of 684.

$$x=\frac{-\left(-6\right)±6\sqrt{19}}{2\times 9}$$

The opposite of -6 is 6.

$$x=\frac{6±6\sqrt{19}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{6±6\sqrt{19}}{18}$$

Now solve the equation x=\frac{6±6\sqrt{19}}{18} when ± is plus. Add 6 to 6\sqrt{19}.

$$x=\frac{6\sqrt{19}+6}{18}$$

Divide 6+6\sqrt{19} by 18.

$$x=\frac{\sqrt{19}+1}{3}$$

Now solve the equation x=\frac{6±6\sqrt{19}}{18} when ± is minus. Subtract 6\sqrt{19} from 6.

$$x=\frac{6-6\sqrt{19}}{18}$$

Divide 6-6\sqrt{19} by 18.

$$x=\frac{1-\sqrt{19}}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1+\sqrt{19}}{3} for x_{1} and \frac{1-\sqrt{19}}{3} for x_{2}.

$$9x^{2}-6x-18=9\left(x-\frac{\sqrt{19}+1}{3}\right)\left(x-\frac{1-\sqrt{19}}{3}\right)$$

Evaluate

$$9x^{2}-6x-18$$