Solve -3x^2+2x+6

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

Factor

$$-3\left(x-\frac{1-\sqrt{19}}{3}\right)\left(x-\frac{\sqrt{19}+1}{3}\right)$$

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.

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

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.

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

Square 2.

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

Multiply -4 times -3.

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

Multiply 12 times 6.

$$x=\frac{-2±\sqrt{4+72}}{2\left(-3\right)}$$

Add 4 to 72.

$$x=\frac{-2±\sqrt{76}}{2\left(-3\right)}$$

Take the square root of 76.

$$x=\frac{-2±2\sqrt{19}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-2±2\sqrt{19}}{-6}$$

Now solve the equation x=\frac{-2±2\sqrt{19}}{-6} when ± is plus. Add -2 to 2\sqrt{19}.

$$x=\frac{2\sqrt{19}-2}{-6}$$

Divide -2+2\sqrt{19} by -6.

$$x=\frac{1-\sqrt{19}}{3}$$

Now solve the equation x=\frac{-2±2\sqrt{19}}{-6} when ± is minus. Subtract 2\sqrt{19} from -2.

$$x=\frac{-2\sqrt{19}-2}{-6}$$

Divide -2-2\sqrt{19} by -6.

$$x=\frac{\sqrt{19}+1}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1-\sqrt{19}}{3} for x_{1} and \frac{1+\sqrt{19}}{3} for x_{2}.

$$-3x^{2}+2x+6=-3\left(x-\frac{1-\sqrt{19}}{3}\right)\left(x-\frac{\sqrt{19}+1}{3}\right)$$

Evaluate

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