Solve (2u^2-6u-3)+(7u^2-5u+1)

$$( 2 u ^ { 2 } - 6 u - 3 ) + ( 7 u ^ { 2 } - 5 u + 1 )$$

Evaluate

$$9u^{2}-11u-2$$

Solution Steps

Combine 2u^{2} and 7u^{2} to get 9u^{2}.

$$9u^{2}-6u-3-5u+1$$

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

$$9u^{2}-11u-3+1$$

Add -3 and 1 to get -2.

$$9u^{2}-11u-2$$

Factor

$$9\left(u-\frac{11-\sqrt{193}}{18}\right)\left(u-\frac{\sqrt{193}+11}{18}\right)$$

Steps Using the Quadratic Formula

Combine 2u^{2} and 7u^{2} to get 9u^{2}.

$$factor(9u^{2}-6u-3-5u+1)$$

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

$$factor(9u^{2}-11u-3+1)$$

Add -3 and 1 to get -2.

$$factor(9u^{2}-11u-2)$$

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.

$$9u^{2}-11u-2=0$$

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.

$$u=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 9\left(-2\right)}}{2\times 9}$$

Square -11.

$$u=\frac{-\left(-11\right)±\sqrt{121-4\times 9\left(-2\right)}}{2\times 9}$$

Multiply -4 times 9.

$$u=\frac{-\left(-11\right)±\sqrt{121-36\left(-2\right)}}{2\times 9}$$

Multiply -36 times -2.

$$u=\frac{-\left(-11\right)±\sqrt{121+72}}{2\times 9}$$

Add 121 to 72.

$$u=\frac{-\left(-11\right)±\sqrt{193}}{2\times 9}$$

The opposite of -11 is 11.

$$u=\frac{11±\sqrt{193}}{2\times 9}$$

Multiply 2 times 9.

$$u=\frac{11±\sqrt{193}}{18}$$

Now solve the equation u=\frac{11±\sqrt{193}}{18} when ± is plus. Add 11 to \sqrt{193}.

$$u=\frac{\sqrt{193}+11}{18}$$

Now solve the equation u=\frac{11±\sqrt{193}}{18} when ± is minus. Subtract \sqrt{193} from 11.

$$u=\frac{11-\sqrt{193}}{18}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{11+\sqrt{193}}{18} for x_{1} and \frac{11-\sqrt{193}}{18} for x_{2}.

$$9u^{2}-11u-2=9\left(u-\frac{\sqrt{193}+11}{18}\right)\left(u-\frac{11-\sqrt{193}}{18}\right)$$