Solve (12x^2+36x-24)

$$( 12 x ^ { 2 } + 36 x - 24 )$$

Factor

$$12\left(x-\frac{-\sqrt{17}-3}{2}\right)\left(x-\frac{\sqrt{17}-3}{2}\right)$$

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.

$$12x^{2}+36x-24=0$$

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.

$$x=\frac{-36±\sqrt{36^{2}-4\times 12\left(-24\right)}}{2\times 12}$$

Square 36.

$$x=\frac{-36±\sqrt{1296-4\times 12\left(-24\right)}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-36±\sqrt{1296-48\left(-24\right)}}{2\times 12}$$

Multiply -48 times -24.

$$x=\frac{-36±\sqrt{1296+1152}}{2\times 12}$$

Add 1296 to 1152.

$$x=\frac{-36±\sqrt{2448}}{2\times 12}$$

Take the square root of 2448.

$$x=\frac{-36±12\sqrt{17}}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{-36±12\sqrt{17}}{24}$$

Now solve the equation x=\frac{-36±12\sqrt{17}}{24} when ± is plus. Add -36 to 12\sqrt{17}.

$$x=\frac{12\sqrt{17}-36}{24}$$

Divide -36+12\sqrt{17} by 24.

$$x=\frac{\sqrt{17}-3}{2}$$

Now solve the equation x=\frac{-36±12\sqrt{17}}{24} when ± is minus. Subtract 12\sqrt{17} from -36.

$$x=\frac{-12\sqrt{17}-36}{24}$$

Divide -36-12\sqrt{17} by 24.

$$x=\frac{-\sqrt{17}-3}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-3+\sqrt{17}}{2} for x_{1} and \frac{-3-\sqrt{17}}{2} for x_{2}.

$$12x^{2}+36x-24=12\left(x-\frac{\sqrt{17}-3}{2}\right)\left(x-\frac{-\sqrt{17}-3}{2}\right)$$

Evaluate

$$12\left(x^{2}+3x-2\right)$$