Solve f(x)=2-8x-6x^2

$$f ( x ) = 2 - 8 x - 6 x ^ { 2 }$$

Factor

$$-6\left(x-\frac{-\sqrt{7}-2}{3}\right)\left(x-\frac{\sqrt{7}-2}{3}\right)$$

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.

$$-6x^{2}-8x+2=0$$

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.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-6\right)\times 2}}{2\left(-6\right)}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-6\right)\times 2}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-\left(-8\right)±\sqrt{64+24\times 2}}{2\left(-6\right)}$$

Multiply 24 times 2.

$$x=\frac{-\left(-8\right)±\sqrt{64+48}}{2\left(-6\right)}$$

Add 64 to 48.

$$x=\frac{-\left(-8\right)±\sqrt{112}}{2\left(-6\right)}$$

Take the square root of 112.

$$x=\frac{-\left(-8\right)±4\sqrt{7}}{2\left(-6\right)}$$

The opposite of -8 is 8.

$$x=\frac{8±4\sqrt{7}}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{8±4\sqrt{7}}{-12}$$

Now solve the equation x=\frac{8±4\sqrt{7}}{-12} when ± is plus. Add 8 to 4\sqrt{7}.

$$x=\frac{4\sqrt{7}+8}{-12}$$

Divide 8+4\sqrt{7} by -12.

$$x=\frac{-\sqrt{7}-2}{3}$$

Now solve the equation x=\frac{8±4\sqrt{7}}{-12} when ± is minus. Subtract 4\sqrt{7} from 8.

$$x=\frac{8-4\sqrt{7}}{-12}$$

Divide 8-4\sqrt{7} by -12.

$$x=\frac{\sqrt{7}-2}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-2-\sqrt{7}}{3} for x_{1} and \frac{-2+\sqrt{7}}{3} for x_{2}.

$$-6x^{2}-8x+2=-6\left(x-\frac{-\sqrt{7}-2}{3}\right)\left(x-\frac{\sqrt{7}-2}{3}\right)$$

Evaluate

$$2-8x-6x^{2}$$