Solve -16x^2+80x+32

$$-16 { x }^{ 2 } +80x+32$$

Factor

$$-16\left(x-\frac{5-\sqrt{33}}{2}\right)\left(x-\frac{\sqrt{33}+5}{2}\right)$$

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.

$$-16x^{2}+80x+32=0$$

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.

$$x=\frac{-80±\sqrt{80^{2}-4\left(-16\right)\times 32}}{2\left(-16\right)}$$

Square 80.

$$x=\frac{-80±\sqrt{6400-4\left(-16\right)\times 32}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-80±\sqrt{6400+64\times 32}}{2\left(-16\right)}$$

Multiply 64 times 32.

$$x=\frac{-80±\sqrt{6400+2048}}{2\left(-16\right)}$$

Add 6400 to 2048.

$$x=\frac{-80±\sqrt{8448}}{2\left(-16\right)}$$

Take the square root of 8448.

$$x=\frac{-80±16\sqrt{33}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{-80±16\sqrt{33}}{-32}$$

Now solve the equation x=\frac{-80±16\sqrt{33}}{-32} when ± is plus. Add -80 to 16\sqrt{33}.

$$x=\frac{16\sqrt{33}-80}{-32}$$

Divide -80+16\sqrt{33} by -32.

$$x=\frac{5-\sqrt{33}}{2}$$

Now solve the equation x=\frac{-80±16\sqrt{33}}{-32} when ± is minus. Subtract 16\sqrt{33} from -80.

$$x=\frac{-16\sqrt{33}-80}{-32}$$

Divide -80-16\sqrt{33} by -32.

$$x=\frac{\sqrt{33}+5}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5-\sqrt{33}}{2} for x_{1} and \frac{5+\sqrt{33}}{2} for x_{2}.

$$-16x^{2}+80x+32=-16\left(x-\frac{5-\sqrt{33}}{2}\right)\left(x-\frac{\sqrt{33}+5}{2}\right)$$

Evaluate

$$32+80x-16x^{2}$$