Solve 8x^2-80x-75

$$8 { x }^{ 2 } -80x-75$$

Factor

$$8\left(x-\left(-\frac{5\sqrt{22}}{4}+5\right)\right)\left(x-\left(\frac{5\sqrt{22}}{4}+5\right)\right)$$

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.

$$8x^{2}-80x-75=0$$

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.

$$x=\frac{-\left(-80\right)±\sqrt{\left(-80\right)^{2}-4\times 8\left(-75\right)}}{2\times 8}$$

Square -80.

$$x=\frac{-\left(-80\right)±\sqrt{6400-4\times 8\left(-75\right)}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-80\right)±\sqrt{6400-32\left(-75\right)}}{2\times 8}$$

Multiply -32 times -75.

$$x=\frac{-\left(-80\right)±\sqrt{6400+2400}}{2\times 8}$$

Add 6400 to 2400.

$$x=\frac{-\left(-80\right)±\sqrt{8800}}{2\times 8}$$

Take the square root of 8800.

$$x=\frac{-\left(-80\right)±20\sqrt{22}}{2\times 8}$$

The opposite of -80 is 80.

$$x=\frac{80±20\sqrt{22}}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{80±20\sqrt{22}}{16}$$

Now solve the equation x=\frac{80±20\sqrt{22}}{16} when ± is plus. Add 80 to 20\sqrt{22}.

$$x=\frac{20\sqrt{22}+80}{16}$$

Divide 80+20\sqrt{22} by 16.

$$x=\frac{5\sqrt{22}}{4}+5$$

Now solve the equation x=\frac{80±20\sqrt{22}}{16} when ± is minus. Subtract 20\sqrt{22} from 80.

$$x=\frac{80-20\sqrt{22}}{16}$$

Divide 80-20\sqrt{22} by 16.

$$x=-\frac{5\sqrt{22}}{4}+5$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 5+\frac{5\sqrt{22}}{4} for x_{1} and 5-\frac{5\sqrt{22}}{4} for x_{2}.

$$8x^{2}-80x-75=8\left(x-\left(\frac{5\sqrt{22}}{4}+5\right)\right)\left(x-\left(-\frac{5\sqrt{22}}{4}+5\right)\right)$$

Evaluate

$$8x^{2}-80x-75$$