Solve 2-2b^2+6b

$$2 - 2 b ^ { 2 } + 6 b$$

Factor

$$-2\left(b-\frac{3-\sqrt{13}}{2}\right)\left(b-\frac{\sqrt{13}+3}{2}\right)$$

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.

$$-2b^{2}+6b+2=0$$

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.

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

Square 6.

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

Multiply -4 times -2.

$$b=\frac{-6±\sqrt{36+8\times 2}}{2\left(-2\right)}$$

Multiply 8 times 2.

$$b=\frac{-6±\sqrt{36+16}}{2\left(-2\right)}$$

Add 36 to 16.

$$b=\frac{-6±\sqrt{52}}{2\left(-2\right)}$$

Take the square root of 52.

$$b=\frac{-6±2\sqrt{13}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$b=\frac{-6±2\sqrt{13}}{-4}$$

Now solve the equation b=\frac{-6±2\sqrt{13}}{-4} when ± is plus. Add -6 to 2\sqrt{13}.

$$b=\frac{2\sqrt{13}-6}{-4}$$

Divide -6+2\sqrt{13} by -4.

$$b=\frac{3-\sqrt{13}}{2}$$

Now solve the equation b=\frac{-6±2\sqrt{13}}{-4} when ± is minus. Subtract 2\sqrt{13} from -6.

$$b=\frac{-2\sqrt{13}-6}{-4}$$

Divide -6-2\sqrt{13} by -4.

$$b=\frac{\sqrt{13}+3}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3-\sqrt{13}}{2} for x_{1} and \frac{3+\sqrt{13}}{2} for x_{2}.

$$-2b^{2}+6b+2=-2\left(b-\frac{3-\sqrt{13}}{2}\right)\left(b-\frac{\sqrt{13}+3}{2}\right)$$

Evaluate

$$2+6b-2b^{2}$$