Solve (11n^2+2n-8)+(4n^2-5n+7)

$$( 11 n ^ { 2 } + 2 n - 8 ) + ( 4 n ^ { 2 } - 5 n + 7 )$$

Evaluate

$$15n^{2}-3n-1$$

Solution Steps

Combine 11n^{2} and 4n^{2} to get 15n^{2}.

$$15n^{2}+2n-8-5n+7$$

Combine 2n and -5n to get -3n.

$$15n^{2}-3n-8+7$$

Add -8 and 7 to get -1.

$$15n^{2}-3n-1$$

Factor

$$15\left(n-\left(-\frac{\sqrt{69}}{30}+\frac{1}{10}\right)\right)\left(n-\left(\frac{\sqrt{69}}{30}+\frac{1}{10}\right)\right)$$

Steps Using the Quadratic Formula

Combine 11n^{2} and 4n^{2} to get 15n^{2}.

$$factor(15n^{2}+2n-8-5n+7)$$

Combine 2n and -5n to get -3n.

$$factor(15n^{2}-3n-8+7)$$

Add -8 and 7 to get -1.

$$factor(15n^{2}-3n-1)$$

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.

$$15n^{2}-3n-1=0$$

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.

$$n=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 15\left(-1\right)}}{2\times 15}$$

Square -3.

$$n=\frac{-\left(-3\right)±\sqrt{9-4\times 15\left(-1\right)}}{2\times 15}$$

Multiply -4 times 15.

$$n=\frac{-\left(-3\right)±\sqrt{9-60\left(-1\right)}}{2\times 15}$$

Multiply -60 times -1.

$$n=\frac{-\left(-3\right)±\sqrt{9+60}}{2\times 15}$$

Add 9 to 60.

$$n=\frac{-\left(-3\right)±\sqrt{69}}{2\times 15}$$

The opposite of -3 is 3.

$$n=\frac{3±\sqrt{69}}{2\times 15}$$

Multiply 2 times 15.

$$n=\frac{3±\sqrt{69}}{30}$$

Now solve the equation n=\frac{3±\sqrt{69}}{30} when ± is plus. Add 3 to \sqrt{69}.

$$n=\frac{\sqrt{69}+3}{30}$$

Divide 3+\sqrt{69} by 30.

$$n=\frac{\sqrt{69}}{30}+\frac{1}{10}$$

Now solve the equation n=\frac{3±\sqrt{69}}{30} when ± is minus. Subtract \sqrt{69} from 3.

$$n=\frac{3-\sqrt{69}}{30}$$

Divide 3-\sqrt{69} by 30.

$$n=-\frac{\sqrt{69}}{30}+\frac{1}{10}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{10}+\frac{\sqrt{69}}{30} for x_{1} and \frac{1}{10}-\frac{\sqrt{69}}{30} for x_{2}.

$$15n^{2}-3n-1=15\left(n-\left(\frac{\sqrt{69}}{30}+\frac{1}{10}\right)\right)\left(n-\left(-\frac{\sqrt{69}}{30}+\frac{1}{10}\right)\right)$$