Solve -3x^2+12x-6

$$-3 { x }^{ 2 } +12x-6$$

Factor

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

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.

$$-3x^{2}+12x-6=0$$

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.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-3\right)\left(-6\right)}}{2\left(-3\right)}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\left(-3\right)\left(-6\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-12±\sqrt{144+12\left(-6\right)}}{2\left(-3\right)}$$

Multiply 12 times -6.

$$x=\frac{-12±\sqrt{144-72}}{2\left(-3\right)}$$

Add 144 to -72.

$$x=\frac{-12±\sqrt{72}}{2\left(-3\right)}$$

Take the square root of 72.

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

Multiply 2 times -3.

$$x=\frac{-12±6\sqrt{2}}{-6}$$

Now solve the equation x=\frac{-12±6\sqrt{2}}{-6} when ± is plus. Add -12 to 6\sqrt{2}.

$$x=\frac{6\sqrt{2}-12}{-6}$$

Divide -12+6\sqrt{2} by -6.

$$x=2-\sqrt{2}$$

Now solve the equation x=\frac{-12±6\sqrt{2}}{-6} when ± is minus. Subtract 6\sqrt{2} from -12.

$$x=\frac{-6\sqrt{2}-12}{-6}$$

Divide -12-6\sqrt{2} by -6.

$$x=\sqrt{2}+2$$

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

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

Evaluate

$$-3x^{2}+12x-6$$