Solve 2x^2+12x-45

$$2 { x }^{ 2 } +12x-45$$

Factor

$$2\left(x-\left(-\frac{3\sqrt{14}}{2}-3\right)\right)\left(x-\left(\frac{3\sqrt{14}}{2}-3\right)\right)$$

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.

$$2x^{2}+12x-45=0$$

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.

$$x=\frac{-12±\sqrt{12^{2}-4\times 2\left(-45\right)}}{2\times 2}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 2\left(-45\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-12±\sqrt{144-8\left(-45\right)}}{2\times 2}$$

Multiply -8 times -45.

$$x=\frac{-12±\sqrt{144+360}}{2\times 2}$$

Add 144 to 360.

$$x=\frac{-12±\sqrt{504}}{2\times 2}$$

Take the square root of 504.

$$x=\frac{-12±6\sqrt{14}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-12±6\sqrt{14}}{4}$$

Now solve the equation x=\frac{-12±6\sqrt{14}}{4} when ± is plus. Add -12 to 6\sqrt{14}.

$$x=\frac{6\sqrt{14}-12}{4}$$

Divide -12+6\sqrt{14} by 4.

$$x=\frac{3\sqrt{14}}{2}-3$$

Now solve the equation x=\frac{-12±6\sqrt{14}}{4} when ± is minus. Subtract 6\sqrt{14} from -12.

$$x=\frac{-6\sqrt{14}-12}{4}$$

Divide -12-6\sqrt{14} by 4.

$$x=-\frac{3\sqrt{14}}{2}-3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -3+\frac{3\sqrt{14}}{2} for x_{1} and -3-\frac{3\sqrt{14}}{2} for x_{2}.

$$2x^{2}+12x-45=2\left(x-\left(\frac{3\sqrt{14}}{2}-3\right)\right)\left(x-\left(-\frac{3\sqrt{14}}{2}-3\right)\right)$$

Evaluate

$$2x^{2}+12x-45$$