Solve (2u^2-6u-7)+(3u^2-5u+4)

$$( 2 u ^ { 2 } - 6 u - 7 ) + ( 3 u ^ { 2 } - 5 u + 4 )$$

Evaluate

$$5u^{2}-11u-3$$

Solution Steps

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

$$5u^{2}-6u-7-5u+4$$

Combine -6u and -5u to get -11u.

$$5u^{2}-11u-7+4$$

Add -7 and 4 to get -3.

$$5u^{2}-11u-3$$

Factor

$$5\left(u-\frac{11-\sqrt{181}}{10}\right)\left(u-\frac{\sqrt{181}+11}{10}\right)$$

Steps Using the Quadratic Formula

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

$$factor(5u^{2}-6u-7-5u+4)$$

Combine -6u and -5u to get -11u.

$$factor(5u^{2}-11u-7+4)$$

Add -7 and 4 to get -3.

$$factor(5u^{2}-11u-3)$$

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.

$$5u^{2}-11u-3=0$$

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.

$$u=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 5\left(-3\right)}}{2\times 5}$$

Square -11.

$$u=\frac{-\left(-11\right)±\sqrt{121-4\times 5\left(-3\right)}}{2\times 5}$$

Multiply -4 times 5.

$$u=\frac{-\left(-11\right)±\sqrt{121-20\left(-3\right)}}{2\times 5}$$

Multiply -20 times -3.

$$u=\frac{-\left(-11\right)±\sqrt{121+60}}{2\times 5}$$

Add 121 to 60.

$$u=\frac{-\left(-11\right)±\sqrt{181}}{2\times 5}$$

The opposite of -11 is 11.

$$u=\frac{11±\sqrt{181}}{2\times 5}$$

Multiply 2 times 5.

$$u=\frac{11±\sqrt{181}}{10}$$

Now solve the equation u=\frac{11±\sqrt{181}}{10} when ± is plus. Add 11 to \sqrt{181}.

$$u=\frac{\sqrt{181}+11}{10}$$

Now solve the equation u=\frac{11±\sqrt{181}}{10} when ± is minus. Subtract \sqrt{181} from 11.

$$u=\frac{11-\sqrt{181}}{10}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{11+\sqrt{181}}{10} for x_{1} and \frac{11-\sqrt{181}}{10} for x_{2}.

$$5u^{2}-11u-3=5\left(u-\frac{\sqrt{181}+11}{10}\right)\left(u-\frac{11-\sqrt{181}}{10}\right)$$