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.

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

Square -6.

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

Multiply -4 times -3.

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

Add 36 to 12.

$$x=\frac{-\left(-6\right)±\sqrt{48}}{2\left(-3\right)}$$

Take the square root of 48.

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

The opposite of -6 is 6.

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

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.

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

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

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