Solve 15x^2-24x-36

$$15 { x }^{ 2 } -24x-36$$

Factor

$$15\left(x-\frac{4-2\sqrt{19}}{5}\right)\left(x-\frac{2\sqrt{19}+4}{5}\right)$$

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.

$$15x^{2}-24x-36=0$$

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.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 15\left(-36\right)}}{2\times 15}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 15\left(-36\right)}}{2\times 15}$$

Multiply -4 times 15.

$$x=\frac{-\left(-24\right)±\sqrt{576-60\left(-36\right)}}{2\times 15}$$

Multiply -60 times -36.

$$x=\frac{-\left(-24\right)±\sqrt{576+2160}}{2\times 15}$$

Add 576 to 2160.

$$x=\frac{-\left(-24\right)±\sqrt{2736}}{2\times 15}$$

Take the square root of 2736.

$$x=\frac{-\left(-24\right)±12\sqrt{19}}{2\times 15}$$

The opposite of -24 is 24.

$$x=\frac{24±12\sqrt{19}}{2\times 15}$$

Multiply 2 times 15.

$$x=\frac{24±12\sqrt{19}}{30}$$

Now solve the equation x=\frac{24±12\sqrt{19}}{30} when ± is plus. Add 24 to 12\sqrt{19}.

$$x=\frac{12\sqrt{19}+24}{30}$$

Divide 24+12\sqrt{19} by 30.

$$x=\frac{2\sqrt{19}+4}{5}$$

Now solve the equation x=\frac{24±12\sqrt{19}}{30} when ± is minus. Subtract 12\sqrt{19} from 24.

$$x=\frac{24-12\sqrt{19}}{30}$$

Divide 24-12\sqrt{19} by 30.

$$x=\frac{4-2\sqrt{19}}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{4+2\sqrt{19}}{5} for x_{1} and \frac{4-2\sqrt{19}}{5} for x_{2}.

$$15x^{2}-24x-36=15\left(x-\frac{2\sqrt{19}+4}{5}\right)\left(x-\frac{4-2\sqrt{19}}{5}\right)$$

Evaluate

$$15x^{2}-24x-36$$