Solve 3x-3x^2+4

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

Factor

$$-3\left(x-\left(-\frac{\sqrt{57}}{6}+\frac{1}{2}\right)\right)\left(x-\left(\frac{\sqrt{57}}{6}+\frac{1}{2}\right)\right)$$

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.

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

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.

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

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-3\right)\times 4}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-3±\sqrt{9+12\times 4}}{2\left(-3\right)}$$

Multiply 12 times 4.

$$x=\frac{-3±\sqrt{9+48}}{2\left(-3\right)}$$

Add 9 to 48.

$$x=\frac{-3±\sqrt{57}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-3±\sqrt{57}}{-6}$$

Now solve the equation x=\frac{-3±\sqrt{57}}{-6} when ± is plus. Add -3 to \sqrt{57}.

$$x=\frac{\sqrt{57}-3}{-6}$$

Divide -3+\sqrt{57} by -6.

$$x=-\frac{\sqrt{57}}{6}+\frac{1}{2}$$

Now solve the equation x=\frac{-3±\sqrt{57}}{-6} when ± is minus. Subtract \sqrt{57} from -3.

$$x=\frac{-\sqrt{57}-3}{-6}$$

Divide -3-\sqrt{57} by -6.

$$x=\frac{\sqrt{57}}{6}+\frac{1}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{2}-\frac{\sqrt{57}}{6} for x_{1} and \frac{1}{2}+\frac{\sqrt{57}}{6} for x_{2}.

$$-3x^{2}+3x+4=-3\left(x-\left(-\frac{\sqrt{57}}{6}+\frac{1}{2}\right)\right)\left(x-\left(\frac{\sqrt{57}}{6}+\frac{1}{2}\right)\right)$$

Evaluate

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