Solve -2a^2-2a+6

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

Factor

$$-2\left(a-\frac{-\sqrt{13}-1}{2}\right)\left(a-\frac{\sqrt{13}-1}{2}\right)$$

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.

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

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.

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

Square -2.

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

Multiply -4 times -2.

$$a=\frac{-\left(-2\right)±\sqrt{4+8\times 6}}{2\left(-2\right)}$$

Multiply 8 times 6.

$$a=\frac{-\left(-2\right)±\sqrt{4+48}}{2\left(-2\right)}$$

Add 4 to 48.

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

Take the square root of 52.

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

The opposite of -2 is 2.

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

Multiply 2 times -2.

$$a=\frac{2±2\sqrt{13}}{-4}$$

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

$$a=\frac{2\sqrt{13}+2}{-4}$$

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

$$a=\frac{-\sqrt{13}-1}{2}$$

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

$$a=\frac{2-2\sqrt{13}}{-4}$$

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

$$a=\frac{\sqrt{13}-1}{2}$$

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

$$-2a^{2}-2a+6=-2\left(a-\frac{-\sqrt{13}-1}{2}\right)\left(a-\frac{\sqrt{13}-1}{2}\right)$$

Evaluate

$$6-2a-2a^{2}$$