Solve -3x^2+6x+4

$$-3 { x }^{ 2 } +6x+4$$

Factor

$$-3\left(x-\left(-\frac{\sqrt{21}}{3}+1\right)\right)\left(x-\left(\frac{\sqrt{21}}{3}+1\right)\right)$$

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.

$$-3x^{2}+6x+4=0$$

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.

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

Square 6.

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

Multiply -4 times -3.

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

Multiply 12 times 4.

$$x=\frac{-6±\sqrt{36+48}}{2\left(-3\right)}$$

Add 36 to 48.

$$x=\frac{-6±\sqrt{84}}{2\left(-3\right)}$$

Take the square root of 84.

$$x=\frac{-6±2\sqrt{21}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-6±2\sqrt{21}}{-6}$$

Now solve the equation x=\frac{-6±2\sqrt{21}}{-6} when ± is plus. Add -6 to 2\sqrt{21}.

$$x=\frac{2\sqrt{21}-6}{-6}$$

Divide -6+2\sqrt{21} by -6.

$$x=-\frac{\sqrt{21}}{3}+1$$

Now solve the equation x=\frac{-6±2\sqrt{21}}{-6} when ± is minus. Subtract 2\sqrt{21} from -6.

$$x=\frac{-2\sqrt{21}-6}{-6}$$

Divide -6-2\sqrt{21} by -6.

$$x=\frac{\sqrt{21}}{3}+1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1-\frac{\sqrt{21}}{3} for x_{1} and 1+\frac{\sqrt{21}}{3} for x_{2}.

$$-3x^{2}+6x+4=-3\left(x-\left(-\frac{\sqrt{21}}{3}+1\right)\right)\left(x-\left(\frac{\sqrt{21}}{3}+1\right)\right)$$

Evaluate

$$4+6x-3x^{2}$$