Solve (3n^2-7n+4)+(4n^2+11n-11)=

$$( 3 n ^ { 2 } - 7 n + 4 ) + ( 4 n ^ { 2 } + 11 n - 11 ) =$$

Evaluate

$$7n^{2}+4n-7$$

Solution Steps

Combine 3n^{2} and 4n^{2} to get 7n^{2}.

$$7n^{2}-7n+4+11n-11$$

Combine -7n and 11n to get 4n.

$$7n^{2}+4n+4-11$$

Subtract 11 from 4 to get -7.

$$7n^{2}+4n-7$$

Factor

$$7\left(n-\frac{-\sqrt{53}-2}{7}\right)\left(n-\frac{\sqrt{53}-2}{7}\right)$$

Steps Using the Quadratic Formula

Combine 3n^{2} and 4n^{2} to get 7n^{2}.

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

Combine -7n and 11n to get 4n.

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

Subtract 11 from 4 to get -7.

$$factor(7n^{2}+4n-7)$$

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.

$$7n^{2}+4n-7=0$$

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.

$$n=\frac{-4±\sqrt{4^{2}-4\times 7\left(-7\right)}}{2\times 7}$$

Square 4.

$$n=\frac{-4±\sqrt{16-4\times 7\left(-7\right)}}{2\times 7}$$

Multiply -4 times 7.

$$n=\frac{-4±\sqrt{16-28\left(-7\right)}}{2\times 7}$$

Multiply -28 times -7.

$$n=\frac{-4±\sqrt{16+196}}{2\times 7}$$

Add 16 to 196.

$$n=\frac{-4±\sqrt{212}}{2\times 7}$$

Take the square root of 212.

$$n=\frac{-4±2\sqrt{53}}{2\times 7}$$

Multiply 2 times 7.

$$n=\frac{-4±2\sqrt{53}}{14}$$

Now solve the equation n=\frac{-4±2\sqrt{53}}{14} when ± is plus. Add -4 to 2\sqrt{53}.

$$n=\frac{2\sqrt{53}-4}{14}$$

Divide -4+2\sqrt{53} by 14.

$$n=\frac{\sqrt{53}-2}{7}$$

Now solve the equation n=\frac{-4±2\sqrt{53}}{14} when ± is minus. Subtract 2\sqrt{53} from -4.

$$n=\frac{-2\sqrt{53}-4}{14}$$

Divide -4-2\sqrt{53} by 14.

$$n=\frac{-\sqrt{53}-2}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-2+\sqrt{53}}{7} for x_{1} and \frac{-2-\sqrt{53}}{7} for x_{2}.

$$7n^{2}+4n-7=7\left(n-\frac{\sqrt{53}-2}{7}\right)\left(n-\frac{-\sqrt{53}-2}{7}\right)$$