Solve 12x^2-15x-5

$$12 { x }^{ 2 } -15x-5$$

Factor

$$12\left(x-\left(-\frac{\sqrt{465}}{24}+\frac{5}{8}\right)\right)\left(x-\left(\frac{\sqrt{465}}{24}+\frac{5}{8}\right)\right)$$

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.

$$12x^{2}-15x-5=0$$

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.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 12\left(-5\right)}}{2\times 12}$$

Square -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times 12\left(-5\right)}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-\left(-15\right)±\sqrt{225-48\left(-5\right)}}{2\times 12}$$

Multiply -48 times -5.

$$x=\frac{-\left(-15\right)±\sqrt{225+240}}{2\times 12}$$

Add 225 to 240.

$$x=\frac{-\left(-15\right)±\sqrt{465}}{2\times 12}$$

The opposite of -15 is 15.

$$x=\frac{15±\sqrt{465}}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{15±\sqrt{465}}{24}$$

Now solve the equation x=\frac{15±\sqrt{465}}{24} when ± is plus. Add 15 to \sqrt{465}.

$$x=\frac{\sqrt{465}+15}{24}$$

Divide 15+\sqrt{465} by 24.

$$x=\frac{\sqrt{465}}{24}+\frac{5}{8}$$

Now solve the equation x=\frac{15±\sqrt{465}}{24} when ± is minus. Subtract \sqrt{465} from 15.

$$x=\frac{15-\sqrt{465}}{24}$$

Divide 15-\sqrt{465} by 24.

$$x=-\frac{\sqrt{465}}{24}+\frac{5}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5}{8}+\frac{\sqrt{465}}{24} for x_{1} and \frac{5}{8}-\frac{\sqrt{465}}{24} for x_{2}.

$$12x^{2}-15x-5=12\left(x-\left(\frac{\sqrt{465}}{24}+\frac{5}{8}\right)\right)\left(x-\left(-\frac{\sqrt{465}}{24}+\frac{5}{8}\right)\right)$$

Evaluate

$$12x^{2}-15x-5$$