Solve 3x^2+6x-1

$$3 { x }^{ 2 } +6x-1$$

Factor

$$3\left(x-\left(-\frac{2\sqrt{3}}{3}-1\right)\right)\left(x-\left(\frac{2\sqrt{3}}{3}-1\right)\right)$$

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.

$$3x^{2}+6x-1=0$$

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.

$$x=\frac{-6±\sqrt{6^{2}-4\times 3\left(-1\right)}}{2\times 3}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\times 3\left(-1\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-6±\sqrt{36-12\left(-1\right)}}{2\times 3}$$

Multiply -12 times -1.

$$x=\frac{-6±\sqrt{36+12}}{2\times 3}$$

Add 36 to 12.

$$x=\frac{-6±\sqrt{48}}{2\times 3}$$

Take the square root of 48.

$$x=\frac{-6±4\sqrt{3}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-6±4\sqrt{3}}{6}$$

Now solve the equation x=\frac{-6±4\sqrt{3}}{6} when ± is plus. Add -6 to 4\sqrt{3}.

$$x=\frac{4\sqrt{3}-6}{6}$$

Divide -6+4\sqrt{3} by 6.

$$x=\frac{2\sqrt{3}}{3}-1$$

Now solve the equation x=\frac{-6±4\sqrt{3}}{6} when ± is minus. Subtract 4\sqrt{3} from -6.

$$x=\frac{-4\sqrt{3}-6}{6}$$

Divide -6-4\sqrt{3} by 6.

$$x=-\frac{2\sqrt{3}}{3}-1$$

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

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

Evaluate

$$3x^{2}+6x-1$$