Solve 2x^2+96x+106

$$2 { x }^{ 2 } +96x+106$$

Factor

$$2\left(x-\left(-\sqrt{523}-24\right)\right)\left(x-\left(\sqrt{523}-24\right)\right)$$

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.

$$2x^{2}+96x+106=0$$

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.

$$x=\frac{-96±\sqrt{96^{2}-4\times 2\times 106}}{2\times 2}$$

Square 96.

$$x=\frac{-96±\sqrt{9216-4\times 2\times 106}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-96±\sqrt{9216-8\times 106}}{2\times 2}$$

Multiply -8 times 106.

$$x=\frac{-96±\sqrt{9216-848}}{2\times 2}$$

Add 9216 to -848.

$$x=\frac{-96±\sqrt{8368}}{2\times 2}$$

Take the square root of 8368.

$$x=\frac{-96±4\sqrt{523}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-96±4\sqrt{523}}{4}$$

Now solve the equation x=\frac{-96±4\sqrt{523}}{4} when ± is plus. Add -96 to 4\sqrt{523}.

$$x=\frac{4\sqrt{523}-96}{4}$$

Divide -96+4\sqrt{523} by 4.

$$x=\sqrt{523}-24$$

Now solve the equation x=\frac{-96±4\sqrt{523}}{4} when ± is minus. Subtract 4\sqrt{523} from -96.

$$x=\frac{-4\sqrt{523}-96}{4}$$

Divide -96-4\sqrt{523} by 4.

$$x=-\sqrt{523}-24$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -24+\sqrt{523} for x_{1} and -24-\sqrt{523} for x_{2}.

$$2x^{2}+96x+106=2\left(x-\left(\sqrt{523}-24\right)\right)\left(x-\left(-\sqrt{523}-24\right)\right)$$

Evaluate

$$2\left(x^{2}+48x+53\right)$$