Solve (3x^2+4x-1)

$$( 3 x ^ { 2 } + 4 x - 1 )$$

Factor

$$3\left(x-\frac{-\sqrt{7}-2}{3}\right)\left(x-\frac{\sqrt{7}-2}{3}\right)$$

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.

$$3x^{2}+4x-1=0$$

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.

$$x=\frac{-4±\sqrt{4^{2}-4\times 3\left(-1\right)}}{2\times 3}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 3\left(-1\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-4±\sqrt{16-12\left(-1\right)}}{2\times 3}$$

Multiply -12 times -1.

$$x=\frac{-4±\sqrt{16+12}}{2\times 3}$$

Add 16 to 12.

$$x=\frac{-4±\sqrt{28}}{2\times 3}$$

Take the square root of 28.

$$x=\frac{-4±2\sqrt{7}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-4±2\sqrt{7}}{6}$$

Now solve the equation x=\frac{-4±2\sqrt{7}}{6} when ± is plus. Add -4 to 2\sqrt{7}.

$$x=\frac{2\sqrt{7}-4}{6}$$

Divide -4+2\sqrt{7} by 6.

$$x=\frac{\sqrt{7}-2}{3}$$

Now solve the equation x=\frac{-4±2\sqrt{7}}{6} when ± is minus. Subtract 2\sqrt{7} from -4.

$$x=\frac{-2\sqrt{7}-4}{6}$$

Divide -4-2\sqrt{7} by 6.

$$x=\frac{-\sqrt{7}-2}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-2+\sqrt{7}}{3} for x_{1} and \frac{-2-\sqrt{7}}{3} for x_{2}.

$$3x^{2}+4x-1=3\left(x-\frac{\sqrt{7}-2}{3}\right)\left(x-\frac{-\sqrt{7}-2}{3}\right)$$

Evaluate

$$3x^{2}+4x-1$$