Solve 3x^2-4x-21

$$3 { x }^{ 2 } -4x-21$$

Factor

$$3\left(x-\frac{2-\sqrt{67}}{3}\right)\left(x-\frac{\sqrt{67}+2}{3}\right)$$

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.

$$3x^{2}-4x-21=0$$

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.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 3\left(-21\right)}}{2\times 3}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 3\left(-21\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-4\right)±\sqrt{16-12\left(-21\right)}}{2\times 3}$$

Multiply -12 times -21.

$$x=\frac{-\left(-4\right)±\sqrt{16+252}}{2\times 3}$$

Add 16 to 252.

$$x=\frac{-\left(-4\right)±\sqrt{268}}{2\times 3}$$

Take the square root of 268.

$$x=\frac{-\left(-4\right)±2\sqrt{67}}{2\times 3}$$

The opposite of -4 is 4.

$$x=\frac{4±2\sqrt{67}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{4±2\sqrt{67}}{6}$$

Now solve the equation x=\frac{4±2\sqrt{67}}{6} when ± is plus. Add 4 to 2\sqrt{67}.

$$x=\frac{2\sqrt{67}+4}{6}$$

Divide 4+2\sqrt{67} by 6.

$$x=\frac{\sqrt{67}+2}{3}$$

Now solve the equation x=\frac{4±2\sqrt{67}}{6} when ± is minus. Subtract 2\sqrt{67} from 4.

$$x=\frac{4-2\sqrt{67}}{6}$$

Divide 4-2\sqrt{67} by 6.

$$x=\frac{2-\sqrt{67}}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{2+\sqrt{67}}{3} for x_{1} and \frac{2-\sqrt{67}}{3} for x_{2}.

$$3x^{2}-4x-21=3\left(x-\frac{\sqrt{67}+2}{3}\right)\left(x-\frac{2-\sqrt{67}}{3}\right)$$

Evaluate

$$3x^{2}-4x-21$$