Solve -3a^2+2a-5--2a^2+a+6

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

Evaluate

$$1+3a-a^{2}$$

Solution Steps

The opposite of -2a^{2} is 2a^{2}.

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

Combine -3a^{2} and 2a^{2} to get -a^{2}.

$$-a^{2}+2a-5+a+6$$

Combine 2a and a to get 3a.

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

Add -5 and 6 to get 1.

$$-a^{2}+3a+1$$

Factor

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

Steps Using the Quadratic Formula

The opposite of -2a^{2} is 2a^{2}.

$$factor(-3a^{2}+2a-5+2a^{2}+a+6)$$

Combine -3a^{2} and 2a^{2} to get -a^{2}.

$$factor(-a^{2}+2a-5+a+6)$$

Combine 2a and a to get 3a.

$$factor(-a^{2}+3a-5+6)$$

Add -5 and 6 to get 1.

$$factor(-a^{2}+3a+1)$$

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.

$$-a^{2}+3a+1=0$$

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.

$$a=\frac{-3±\sqrt{3^{2}-4\left(-1\right)}}{2\left(-1\right)}$$

Square 3.

$$a=\frac{-3±\sqrt{9-4\left(-1\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$a=\frac{-3±\sqrt{9+4}}{2\left(-1\right)}$$

Add 9 to 4.

$$a=\frac{-3±\sqrt{13}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$a=\frac{-3±\sqrt{13}}{-2}$$

Now solve the equation a=\frac{-3±\sqrt{13}}{-2} when ± is plus. Add -3 to \sqrt{13}.

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

Divide -3+\sqrt{13} by -2.

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

Now solve the equation a=\frac{-3±\sqrt{13}}{-2} when ± is minus. Subtract \sqrt{13} from -3.

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

Divide -3-\sqrt{13} by -2.

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

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}.

$$-a^{2}+3a+1=-\left(a-\frac{3-\sqrt{13}}{2}\right)\left(a-\frac{\sqrt{13}+3}{2}\right)$$