Solve 12+6x-3x^2

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

Factor

$$-3\left(x-\left(1-\sqrt{5}\right)\right)\left(x-\left(\sqrt{5}+1\right)\right)$$

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.

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

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.

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

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-3\right)\times 12}}{2\left(-3\right)}$$

Multiply -4 times -3.

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

Multiply 12 times 12.

$$x=\frac{-6±\sqrt{36+144}}{2\left(-3\right)}$$

Add 36 to 144.

$$x=\frac{-6±\sqrt{180}}{2\left(-3\right)}$$

Take the square root of 180.

$$x=\frac{-6±6\sqrt{5}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-6±6\sqrt{5}}{-6}$$

Now solve the equation x=\frac{-6±6\sqrt{5}}{-6} when ± is plus. Add -6 to 6\sqrt{5}.

$$x=\frac{6\sqrt{5}-6}{-6}$$

Divide -6+6\sqrt{5} by -6.

$$x=1-\sqrt{5}$$

Now solve the equation x=\frac{-6±6\sqrt{5}}{-6} when ± is minus. Subtract 6\sqrt{5} from -6.

$$x=\frac{-6\sqrt{5}-6}{-6}$$

Divide -6-6\sqrt{5} by -6.

$$x=\sqrt{5}+1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1-\sqrt{5} for x_{1} and 1+\sqrt{5} for x_{2}.

$$-3x^{2}+6x+12=-3\left(x-\left(1-\sqrt{5}\right)\right)\left(x-\left(\sqrt{5}+1\right)\right)$$

Evaluate

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