Solve 9x^2-4x-24

$$9 { x }^{ 2 } -4x-24$$

Factor

$$9\left(x-\frac{2-2\sqrt{55}}{9}\right)\left(x-\frac{2\sqrt{55}+2}{9}\right)$$

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.

$$9x^{2}-4x-24=0$$

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.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 9\left(-24\right)}}{2\times 9}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 9\left(-24\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-4\right)±\sqrt{16-36\left(-24\right)}}{2\times 9}$$

Multiply -36 times -24.

$$x=\frac{-\left(-4\right)±\sqrt{16+864}}{2\times 9}$$

Add 16 to 864.

$$x=\frac{-\left(-4\right)±\sqrt{880}}{2\times 9}$$

Take the square root of 880.

$$x=\frac{-\left(-4\right)±4\sqrt{55}}{2\times 9}$$

The opposite of -4 is 4.

$$x=\frac{4±4\sqrt{55}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{4±4\sqrt{55}}{18}$$

Now solve the equation x=\frac{4±4\sqrt{55}}{18} when ± is plus. Add 4 to 4\sqrt{55}.

$$x=\frac{4\sqrt{55}+4}{18}$$

Divide 4+4\sqrt{55} by 18.

$$x=\frac{2\sqrt{55}+2}{9}$$

Now solve the equation x=\frac{4±4\sqrt{55}}{18} when ± is minus. Subtract 4\sqrt{55} from 4.

$$x=\frac{4-4\sqrt{55}}{18}$$

Divide 4-4\sqrt{55} by 18.

$$x=\frac{2-2\sqrt{55}}{9}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{2+2\sqrt{55}}{9} for x_{1} and \frac{2-2\sqrt{55}}{9} for x_{2}.

$$9x^{2}-4x-24=9\left(x-\frac{2\sqrt{55}+2}{9}\right)\left(x-\frac{2-2\sqrt{55}}{9}\right)$$

Evaluate

$$9x^{2}-4x-24$$