Solve 16x^2-24x-11

$$16 { x }^{ 2 } -24x-11$$

Factor

$$16\left(x-\left(-\frac{\sqrt{5}}{2}+\frac{3}{4}\right)\right)\left(x-\left(\frac{\sqrt{5}}{2}+\frac{3}{4}\right)\right)$$

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.

$$16x^{2}-24x-11=0$$

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.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 16\left(-11\right)}}{2\times 16}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 16\left(-11\right)}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-\left(-24\right)±\sqrt{576-64\left(-11\right)}}{2\times 16}$$

Multiply -64 times -11.

$$x=\frac{-\left(-24\right)±\sqrt{576+704}}{2\times 16}$$

Add 576 to 704.

$$x=\frac{-\left(-24\right)±\sqrt{1280}}{2\times 16}$$

Take the square root of 1280.

$$x=\frac{-\left(-24\right)±16\sqrt{5}}{2\times 16}$$

The opposite of -24 is 24.

$$x=\frac{24±16\sqrt{5}}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{24±16\sqrt{5}}{32}$$

Now solve the equation x=\frac{24±16\sqrt{5}}{32} when ± is plus. Add 24 to 16\sqrt{5}.

$$x=\frac{16\sqrt{5}+24}{32}$$

Divide 24+16\sqrt{5} by 32.

$$x=\frac{\sqrt{5}}{2}+\frac{3}{4}$$

Now solve the equation x=\frac{24±16\sqrt{5}}{32} when ± is minus. Subtract 16\sqrt{5} from 24.

$$x=\frac{24-16\sqrt{5}}{32}$$

Divide 24-16\sqrt{5} by 32.

$$x=-\frac{\sqrt{5}}{2}+\frac{3}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3}{4}+\frac{\sqrt{5}}{2} for x_{1} and \frac{3}{4}-\frac{\sqrt{5}}{2} for x_{2}.

$$16x^{2}-24x-11=16\left(x-\left(\frac{\sqrt{5}}{2}+\frac{3}{4}\right)\right)\left(x-\left(-\frac{\sqrt{5}}{2}+\frac{3}{4}\right)\right)$$

Evaluate

$$16x^{2}-24x-11$$