Solve 3x^2-19x-18

$$3 { x }^{ 2 } -19x-18$$

Factor

$$3\left(x-\frac{19-\sqrt{577}}{6}\right)\left(x-\frac{\sqrt{577}+19}{6}\right)$$

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.

$$3x^{2}-19x-18=0$$

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.

$$x=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\times 3\left(-18\right)}}{2\times 3}$$

Square -19.

$$x=\frac{-\left(-19\right)±\sqrt{361-4\times 3\left(-18\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-19\right)±\sqrt{361-12\left(-18\right)}}{2\times 3}$$

Multiply -12 times -18.

$$x=\frac{-\left(-19\right)±\sqrt{361+216}}{2\times 3}$$

Add 361 to 216.

$$x=\frac{-\left(-19\right)±\sqrt{577}}{2\times 3}$$

The opposite of -19 is 19.

$$x=\frac{19±\sqrt{577}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{19±\sqrt{577}}{6}$$

Now solve the equation x=\frac{19±\sqrt{577}}{6} when ± is plus. Add 19 to \sqrt{577}.

$$x=\frac{\sqrt{577}+19}{6}$$

Now solve the equation x=\frac{19±\sqrt{577}}{6} when ± is minus. Subtract \sqrt{577} from 19.

$$x=\frac{19-\sqrt{577}}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{19+\sqrt{577}}{6} for x_{1} and \frac{19-\sqrt{577}}{6} for x_{2}.

$$3x^{2}-19x-18=3\left(x-\frac{\sqrt{577}+19}{6}\right)\left(x-\frac{19-\sqrt{577}}{6}\right)$$

Evaluate

$$3x^{2}-19x-18$$