Solve 7a+3a^2+2a-a^2+3

$$7 a + 3 a ^ { 2 } + 2 a - a ^ { 2 } + 3$$

Evaluate

$$2a^{2}+9a+3$$

Solution Steps

Combine 7a and 2a to get 9a.

$$9a+3a^{2}-a^{2}+3$$

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

$$9a+2a^{2}+3$$

Factor

$$2\left(a-\frac{-\sqrt{57}-9}{4}\right)\left(a-\frac{\sqrt{57}-9}{4}\right)$$

Steps Using the Quadratic Formula

Combine 7a and 2a to get 9a.

$$factor(9a+3a^{2}-a^{2}+3)$$

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

$$factor(9a+2a^{2}+3)$$

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.

$$2a^{2}+9a+3=0$$

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.

$$a=\frac{-9±\sqrt{9^{2}-4\times 2\times 3}}{2\times 2}$$

Square 9.

$$a=\frac{-9±\sqrt{81-4\times 2\times 3}}{2\times 2}$$

Multiply -4 times 2.

$$a=\frac{-9±\sqrt{81-8\times 3}}{2\times 2}$$

Multiply -8 times 3.

$$a=\frac{-9±\sqrt{81-24}}{2\times 2}$$

Add 81 to -24.

$$a=\frac{-9±\sqrt{57}}{2\times 2}$$

Multiply 2 times 2.

$$a=\frac{-9±\sqrt{57}}{4}$$

Now solve the equation a=\frac{-9±\sqrt{57}}{4} when ± is plus. Add -9 to \sqrt{57}.

$$a=\frac{\sqrt{57}-9}{4}$$

Now solve the equation a=\frac{-9±\sqrt{57}}{4} when ± is minus. Subtract \sqrt{57} from -9.

$$a=\frac{-\sqrt{57}-9}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-9+\sqrt{57}}{4} for x_{1} and \frac{-9-\sqrt{57}}{4} for x_{2}.

$$2a^{2}+9a+3=2\left(a-\frac{\sqrt{57}-9}{4}\right)\left(a-\frac{-\sqrt{57}-9}{4}\right)$$