Solve (5a-3a^2)+(8+7a)

$$( 5 a - 3 a ^ { 2 } ) + ( 8 + 7 a )$$

Factor

$$-3\left(a-\left(-\frac{2\sqrt{15}}{3}+2\right)\right)\left(a-\left(\frac{2\sqrt{15}}{3}+2\right)\right)$$

Steps Using the Quadratic Formula

Combine 5a and 7a to get 12a.

$$factor(12a-3a^{2}+8)$$

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.

$$-3a^{2}+12a+8=0$$

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.

$$a=\frac{-12±\sqrt{12^{2}-4\left(-3\right)\times 8}}{2\left(-3\right)}$$

Square 12.

$$a=\frac{-12±\sqrt{144-4\left(-3\right)\times 8}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$a=\frac{-12±\sqrt{144+12\times 8}}{2\left(-3\right)}$$

Multiply 12 times 8.

$$a=\frac{-12±\sqrt{144+96}}{2\left(-3\right)}$$

Add 144 to 96.

$$a=\frac{-12±\sqrt{240}}{2\left(-3\right)}$$

Take the square root of 240.

$$a=\frac{-12±4\sqrt{15}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$a=\frac{-12±4\sqrt{15}}{-6}$$

Now solve the equation a=\frac{-12±4\sqrt{15}}{-6} when ± is plus. Add -12 to 4\sqrt{15}.

$$a=\frac{4\sqrt{15}-12}{-6}$$

Divide -12+4\sqrt{15} by -6.

$$a=-\frac{2\sqrt{15}}{3}+2$$

Now solve the equation a=\frac{-12±4\sqrt{15}}{-6} when ± is minus. Subtract 4\sqrt{15} from -12.

$$a=\frac{-4\sqrt{15}-12}{-6}$$

Divide -12-4\sqrt{15} by -6.

$$a=\frac{2\sqrt{15}}{3}+2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 2-\frac{2\sqrt{15}}{3} for x_{1} and 2+\frac{2\sqrt{15}}{3} for x_{2}.

$$-3a^{2}+12a+8=-3\left(a-\left(-\frac{2\sqrt{15}}{3}+2\right)\right)\left(a-\left(\frac{2\sqrt{15}}{3}+2\right)\right)$$

Evaluate

$$8+12a-3a^{2}$$

Solution Steps

Combine 5a and 7a to get 12a.

$$12a-3a^{2}+8$$