Solve f(x)=5-3x^2+6x

$$f ( x ) = 5 - 3 x ^ { 2 } + 6 x$$

Factor

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

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.

$$-3x^{2}+6x+5=0$$

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.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-3\right)\times 5}}{2\left(-3\right)}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-3\right)\times 5}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-6±\sqrt{36+12\times 5}}{2\left(-3\right)}$$

Multiply 12 times 5.

$$x=\frac{-6±\sqrt{36+60}}{2\left(-3\right)}$$

Add 36 to 60.

$$x=\frac{-6±\sqrt{96}}{2\left(-3\right)}$$

Take the square root of 96.

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

Multiply 2 times -3.

$$x=\frac{-6±4\sqrt{6}}{-6}$$

Now solve the equation x=\frac{-6±4\sqrt{6}}{-6} when ± is plus. Add -6 to 4\sqrt{6}.

$$x=\frac{4\sqrt{6}-6}{-6}$$

Divide -6+4\sqrt{6} by -6.

$$x=-\frac{2\sqrt{6}}{3}+1$$

Now solve the equation x=\frac{-6±4\sqrt{6}}{-6} when ± is minus. Subtract 4\sqrt{6} from -6.

$$x=\frac{-4\sqrt{6}-6}{-6}$$

Divide -6-4\sqrt{6} by -6.

$$x=\frac{2\sqrt{6}}{3}+1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1-\frac{2\sqrt{6}}{3} for x_{1} and 1+\frac{2\sqrt{6}}{3} for x_{2}.

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

Evaluate

$$5+6x-3x^{2}$$