Solve 6x^2+4x-24

$$6 { x }^{ 2 } +4x-24$$

Factor

$$6\left(x-\frac{-\sqrt{37}-1}{3}\right)\left(x-\frac{\sqrt{37}-1}{3}\right)$$

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.

$$6x^{2}+4x-24=0$$

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.

$$x=\frac{-4±\sqrt{4^{2}-4\times 6\left(-24\right)}}{2\times 6}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 6\left(-24\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-4±\sqrt{16-24\left(-24\right)}}{2\times 6}$$

Multiply -24 times -24.

$$x=\frac{-4±\sqrt{16+576}}{2\times 6}$$

Add 16 to 576.

$$x=\frac{-4±\sqrt{592}}{2\times 6}$$

Take the square root of 592.

$$x=\frac{-4±4\sqrt{37}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-4±4\sqrt{37}}{12}$$

Now solve the equation x=\frac{-4±4\sqrt{37}}{12} when ± is plus. Add -4 to 4\sqrt{37}.

$$x=\frac{4\sqrt{37}-4}{12}$$

Divide -4+4\sqrt{37} by 12.

$$x=\frac{\sqrt{37}-1}{3}$$

Now solve the equation x=\frac{-4±4\sqrt{37}}{12} when ± is minus. Subtract 4\sqrt{37} from -4.

$$x=\frac{-4\sqrt{37}-4}{12}$$

Divide -4-4\sqrt{37} by 12.

$$x=\frac{-\sqrt{37}-1}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-1+\sqrt{37}}{3} for x_{1} and \frac{-1-\sqrt{37}}{3} for x_{2}.

$$6x^{2}+4x-24=6\left(x-\frac{\sqrt{37}-1}{3}\right)\left(x-\frac{-\sqrt{37}-1}{3}\right)$$

Evaluate

$$6x^{2}+4x-24$$