Solve {l}{(-1a^2+2a+3)}{+(3a-4a+5)}

$$\left. \begin{array} { l } { ( - 1 a ^ { 2 } + 2 a + 3 ) } \\ { + ( 3 a - 4 a + 5 ) } \end{array} \right.$$

Evaluate

$$8+a-a^{2}$$

Solution Steps

Combine 2a and 3a to get 5a.

$$-a^{2}+5a+3-4a+5$$

Combine 5a and -4a to get a.

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

Add 3 and 5 to get 8.

$$-a^{2}+a+8$$

Factor

$$-\left(a-\frac{1-\sqrt{33}}{2}\right)\left(a-\frac{\sqrt{33}+1}{2}\right)$$

Steps Using the Quadratic Formula

Combine 2a and 3a to get 5a.

$$factor(-a^{2}+5a+3-4a+5)$$

Combine 5a and -4a to get a.

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

Add 3 and 5 to get 8.

$$factor(-a^{2}+a+8)$$

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.

$$-a^{2}+a+8=0$$

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.

$$a=\frac{-1±\sqrt{1^{2}-4\left(-1\right)\times 8}}{2\left(-1\right)}$$

Square 1.

$$a=\frac{-1±\sqrt{1-4\left(-1\right)\times 8}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$a=\frac{-1±\sqrt{1+4\times 8}}{2\left(-1\right)}$$

Multiply 4 times 8.

$$a=\frac{-1±\sqrt{1+32}}{2\left(-1\right)}$$

Add 1 to 32.

$$a=\frac{-1±\sqrt{33}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$a=\frac{-1±\sqrt{33}}{-2}$$

Now solve the equation a=\frac{-1±\sqrt{33}}{-2} when ± is plus. Add -1 to \sqrt{33}.

$$a=\frac{\sqrt{33}-1}{-2}$$

Divide -1+\sqrt{33} by -2.

$$a=\frac{1-\sqrt{33}}{2}$$

Now solve the equation a=\frac{-1±\sqrt{33}}{-2} when ± is minus. Subtract \sqrt{33} from -1.

$$a=\frac{-\sqrt{33}-1}{-2}$$

Divide -1-\sqrt{33} by -2.

$$a=\frac{\sqrt{33}+1}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1-\sqrt{33}}{2} for x_{1} and \frac{1+\sqrt{33}}{2} for x_{2}.

$$-a^{2}+a+8=-\left(a-\frac{1-\sqrt{33}}{2}\right)\left(a-\frac{\sqrt{33}+1}{2}\right)$$