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

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

Evaluate

$$7n^{2}-6n-7$$

Solution Steps

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

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

Combine -7n and n to get -6n.

$$7n^{2}-6n+4-11$$

Subtract 11 from 4 to get -7.

$$7n^{2}-6n-7$$

Factor

$$7\left(n-\frac{3-\sqrt{58}}{7}\right)\left(n-\frac{\sqrt{58}+3}{7}\right)$$

Steps Using the Quadratic Formula

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

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

Combine -7n and n to get -6n.

$$factor(7n^{2}-6n+4-11)$$

Subtract 11 from 4 to get -7.

$$factor(7n^{2}-6n-7)$$

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.

$$7n^{2}-6n-7=0$$

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.

$$n=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 7\left(-7\right)}}{2\times 7}$$

Square -6.

$$n=\frac{-\left(-6\right)±\sqrt{36-4\times 7\left(-7\right)}}{2\times 7}$$

Multiply -4 times 7.

$$n=\frac{-\left(-6\right)±\sqrt{36-28\left(-7\right)}}{2\times 7}$$

Multiply -28 times -7.

$$n=\frac{-\left(-6\right)±\sqrt{36+196}}{2\times 7}$$

Add 36 to 196.

$$n=\frac{-\left(-6\right)±\sqrt{232}}{2\times 7}$$

Take the square root of 232.

$$n=\frac{-\left(-6\right)±2\sqrt{58}}{2\times 7}$$

The opposite of -6 is 6.

$$n=\frac{6±2\sqrt{58}}{2\times 7}$$

Multiply 2 times 7.

$$n=\frac{6±2\sqrt{58}}{14}$$

Now solve the equation n=\frac{6±2\sqrt{58}}{14} when ± is plus. Add 6 to 2\sqrt{58}.

$$n=\frac{2\sqrt{58}+6}{14}$$

Divide 6+2\sqrt{58} by 14.

$$n=\frac{\sqrt{58}+3}{7}$$

Now solve the equation n=\frac{6±2\sqrt{58}}{14} when ± is minus. Subtract 2\sqrt{58} from 6.

$$n=\frac{6-2\sqrt{58}}{14}$$

Divide 6-2\sqrt{58} by 14.

$$n=\frac{3-\sqrt{58}}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3+\sqrt{58}}{7} for x_{1} and \frac{3-\sqrt{58}}{7} for x_{2}.

$$7n^{2}-6n-7=7\left(n-\frac{\sqrt{58}+3}{7}\right)\left(n-\frac{3-\sqrt{58}}{7}\right)$$