Solve 8x^2-12x-16=

$$8 { x }^{ 2 } -12x-16=$$

Factor

$$8\left(x-\frac{3-\sqrt{41}}{4}\right)\left(x-\frac{\sqrt{41}+3}{4}\right)$$

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.

$$8x^{2}-12x-16=0$$

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.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 8\left(-16\right)}}{2\times 8}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 8\left(-16\right)}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-12\right)±\sqrt{144-32\left(-16\right)}}{2\times 8}$$

Multiply -32 times -16.

$$x=\frac{-\left(-12\right)±\sqrt{144+512}}{2\times 8}$$

Add 144 to 512.

$$x=\frac{-\left(-12\right)±\sqrt{656}}{2\times 8}$$

Take the square root of 656.

$$x=\frac{-\left(-12\right)±4\sqrt{41}}{2\times 8}$$

The opposite of -12 is 12.

$$x=\frac{12±4\sqrt{41}}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{12±4\sqrt{41}}{16}$$

Now solve the equation x=\frac{12±4\sqrt{41}}{16} when ± is plus. Add 12 to 4\sqrt{41}.

$$x=\frac{4\sqrt{41}+12}{16}$$

Divide 12+4\sqrt{41} by 16.

$$x=\frac{\sqrt{41}+3}{4}$$

Now solve the equation x=\frac{12±4\sqrt{41}}{16} when ± is minus. Subtract 4\sqrt{41} from 12.

$$x=\frac{12-4\sqrt{41}}{16}$$

Divide 12-4\sqrt{41} by 16.

$$x=\frac{3-\sqrt{41}}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3+\sqrt{41}}{4} for x_{1} and \frac{3-\sqrt{41}}{4} for x_{2}.

$$8x^{2}-12x-16=8\left(x-\frac{\sqrt{41}+3}{4}\right)\left(x-\frac{3-\sqrt{41}}{4}\right)$$

Evaluate

$$8x^{2}-12x-16$$