Solve 2a^2+8a-13+3a^2-11a-5

$$2 a ^ { 2 } + 8 a - 13 + 3 a ^ { 2 } - 11 a - 5$$

Evaluate

$$5a^{2}-3a-18$$

Solution Steps

Combine 2a^{2} and 3a^{2} to get 5a^{2}.

$$5a^{2}+8a-13-11a-5$$

Combine 8a and -11a to get -3a.

$$5a^{2}-3a-13-5$$

Subtract 5 from -13 to get -18.

$$5a^{2}-3a-18$$

Factor

$$5\left(a-\frac{3-3\sqrt{41}}{10}\right)\left(a-\frac{3\sqrt{41}+3}{10}\right)$$

Steps Using the Quadratic Formula

Combine 2a^{2} and 3a^{2} to get 5a^{2}.

$$factor(5a^{2}+8a-13-11a-5)$$

Combine 8a and -11a to get -3a.

$$factor(5a^{2}-3a-13-5)$$

Subtract 5 from -13 to get -18.

$$factor(5a^{2}-3a-18)$$

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.

$$5a^{2}-3a-18=0$$

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.

$$a=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 5\left(-18\right)}}{2\times 5}$$

Square -3.

$$a=\frac{-\left(-3\right)±\sqrt{9-4\times 5\left(-18\right)}}{2\times 5}$$

Multiply -4 times 5.

$$a=\frac{-\left(-3\right)±\sqrt{9-20\left(-18\right)}}{2\times 5}$$

Multiply -20 times -18.

$$a=\frac{-\left(-3\right)±\sqrt{9+360}}{2\times 5}$$

Add 9 to 360.

$$a=\frac{-\left(-3\right)±\sqrt{369}}{2\times 5}$$

Take the square root of 369.

$$a=\frac{-\left(-3\right)±3\sqrt{41}}{2\times 5}$$

The opposite of -3 is 3.

$$a=\frac{3±3\sqrt{41}}{2\times 5}$$

Multiply 2 times 5.

$$a=\frac{3±3\sqrt{41}}{10}$$

Now solve the equation a=\frac{3±3\sqrt{41}}{10} when ± is plus. Add 3 to 3\sqrt{41}.

$$a=\frac{3\sqrt{41}+3}{10}$$

Now solve the equation a=\frac{3±3\sqrt{41}}{10} when ± is minus. Subtract 3\sqrt{41} from 3.

$$a=\frac{3-3\sqrt{41}}{10}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3+3\sqrt{41}}{10} for x_{1} and \frac{3-3\sqrt{41}}{10} for x_{2}.

$$5a^{2}-3a-18=5\left(a-\frac{3\sqrt{41}+3}{10}\right)\left(a-\frac{3-3\sqrt{41}}{10}\right)$$