Solve 3x^2-5x-9

$$3 { x }^{ 2 } -5x-9$$

Factor

$$3\left(x-\frac{5-\sqrt{133}}{6}\right)\left(x-\frac{\sqrt{133}+5}{6}\right)$$

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.

$$3x^{2}-5x-9=0$$

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.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 3\left(-9\right)}}{2\times 3}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 3\left(-9\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-5\right)±\sqrt{25-12\left(-9\right)}}{2\times 3}$$

Multiply -12 times -9.

$$x=\frac{-\left(-5\right)±\sqrt{25+108}}{2\times 3}$$

Add 25 to 108.

$$x=\frac{-\left(-5\right)±\sqrt{133}}{2\times 3}$$

The opposite of -5 is 5.

$$x=\frac{5±\sqrt{133}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{5±\sqrt{133}}{6}$$

Now solve the equation x=\frac{5±\sqrt{133}}{6} when ± is plus. Add 5 to \sqrt{133}.

$$x=\frac{\sqrt{133}+5}{6}$$

Now solve the equation x=\frac{5±\sqrt{133}}{6} when ± is minus. Subtract \sqrt{133} from 5.

$$x=\frac{5-\sqrt{133}}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5+\sqrt{133}}{6} for x_{1} and \frac{5-\sqrt{133}}{6} for x_{2}.

$$3x^{2}-5x-9=3\left(x-\frac{\sqrt{133}+5}{6}\right)\left(x-\frac{5-\sqrt{133}}{6}\right)$$

Evaluate

$$3x^{2}-5x-9$$