Solve 8x^2-22x-15

$$8 { x }^{ 2 } -22x-15$$

Factor

$$8\left(x-\frac{11-\sqrt{241}}{8}\right)\left(x-\frac{\sqrt{241}+11}{8}\right)$$

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.

$$8x^{2}-22x-15=0$$

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.

$$x=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\times 8\left(-15\right)}}{2\times 8}$$

Square -22.

$$x=\frac{-\left(-22\right)±\sqrt{484-4\times 8\left(-15\right)}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-22\right)±\sqrt{484-32\left(-15\right)}}{2\times 8}$$

Multiply -32 times -15.

$$x=\frac{-\left(-22\right)±\sqrt{484+480}}{2\times 8}$$

Add 484 to 480.

$$x=\frac{-\left(-22\right)±\sqrt{964}}{2\times 8}$$

Take the square root of 964.

$$x=\frac{-\left(-22\right)±2\sqrt{241}}{2\times 8}$$

The opposite of -22 is 22.

$$x=\frac{22±2\sqrt{241}}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{22±2\sqrt{241}}{16}$$

Now solve the equation x=\frac{22±2\sqrt{241}}{16} when ± is plus. Add 22 to 2\sqrt{241}.

$$x=\frac{2\sqrt{241}+22}{16}$$

Divide 22+2\sqrt{241} by 16.

$$x=\frac{\sqrt{241}+11}{8}$$

Now solve the equation x=\frac{22±2\sqrt{241}}{16} when ± is minus. Subtract 2\sqrt{241} from 22.

$$x=\frac{22-2\sqrt{241}}{16}$$

Divide 22-2\sqrt{241} by 16.

$$x=\frac{11-\sqrt{241}}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{11+\sqrt{241}}{8} for x_{1} and \frac{11-\sqrt{241}}{8} for x_{2}.

$$8x^{2}-22x-15=8\left(x-\frac{\sqrt{241}+11}{8}\right)\left(x-\frac{11-\sqrt{241}}{8}\right)$$

Evaluate

$$8x^{2}-22x-15$$