Solve 4+12x-2x^2

$$4+12x-2 { x }^{ 2 }$$

Factor

$$-2\left(x-\left(3-\sqrt{11}\right)\right)\left(x-\left(\sqrt{11}+3\right)\right)$$

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.

$$-2x^{2}+12x+4=0$$

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.

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

Square 12.

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

Multiply -4 times -2.

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

Multiply 8 times 4.

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

Add 144 to 32.

$$x=\frac{-12±\sqrt{176}}{2\left(-2\right)}$$

Take the square root of 176.

$$x=\frac{-12±4\sqrt{11}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-12±4\sqrt{11}}{-4}$$

Now solve the equation x=\frac{-12±4\sqrt{11}}{-4} when ± is plus. Add -12 to 4\sqrt{11}.

$$x=\frac{4\sqrt{11}-12}{-4}$$

Divide -12+4\sqrt{11} by -4.

$$x=3-\sqrt{11}$$

Now solve the equation x=\frac{-12±4\sqrt{11}}{-4} when ± is minus. Subtract 4\sqrt{11} from -12.

$$x=\frac{-4\sqrt{11}-12}{-4}$$

Divide -12-4\sqrt{11} by -4.

$$x=\sqrt{11}+3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 3-\sqrt{11} for x_{1} and 3+\sqrt{11} for x_{2}.

$$-2x^{2}+12x+4=-2\left(x-\left(3-\sqrt{11}\right)\right)\left(x-\left(\sqrt{11}+3\right)\right)$$

Evaluate

$$4+12x-2x^{2}$$