Solve 3x^2+2x-14

$$3 { x }^{ 2 } +2x-14$$

Factor

$$3\left(x-\frac{-\sqrt{43}-1}{3}\right)\left(x-\frac{\sqrt{43}-1}{3}\right)$$

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.

$$3x^{2}+2x-14=0$$

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.

$$x=\frac{-2±\sqrt{2^{2}-4\times 3\left(-14\right)}}{2\times 3}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\times 3\left(-14\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-2±\sqrt{4-12\left(-14\right)}}{2\times 3}$$

Multiply -12 times -14.

$$x=\frac{-2±\sqrt{4+168}}{2\times 3}$$

Add 4 to 168.

$$x=\frac{-2±\sqrt{172}}{2\times 3}$$

Take the square root of 172.

$$x=\frac{-2±2\sqrt{43}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-2±2\sqrt{43}}{6}$$

Now solve the equation x=\frac{-2±2\sqrt{43}}{6} when ± is plus. Add -2 to 2\sqrt{43}.

$$x=\frac{2\sqrt{43}-2}{6}$$

Divide -2+2\sqrt{43} by 6.

$$x=\frac{\sqrt{43}-1}{3}$$

Now solve the equation x=\frac{-2±2\sqrt{43}}{6} when ± is minus. Subtract 2\sqrt{43} from -2.

$$x=\frac{-2\sqrt{43}-2}{6}$$

Divide -2-2\sqrt{43} by 6.

$$x=\frac{-\sqrt{43}-1}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-1+\sqrt{43}}{3} for x_{1} and \frac{-1-\sqrt{43}}{3} for x_{2}.

$$3x^{2}+2x-14=3\left(x-\frac{\sqrt{43}-1}{3}\right)\left(x-\frac{-\sqrt{43}-1}{3}\right)$$

Evaluate

$$3x^{2}+2x-14$$