Solve (-6u^2+5u-3)+(4u^2+7u+1)

$$( - 6 u ^ { 2 } + 5 u - 3 ) + ( 4 u ^ { 2 } + 7 u + 1 )$$

Evaluate

$$-2u^{2}+12u-2$$

Solution Steps

Combine -6u^{2} and 4u^{2} to get -2u^{2}.

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

Combine 5u and 7u to get 12u.

$$-2u^{2}+12u-3+1$$

Add -3 and 1 to get -2.

$$-2u^{2}+12u-2$$

Factor

$$-2\left(u-\left(3-2\sqrt{2}\right)\right)\left(u-\left(2\sqrt{2}+3\right)\right)$$

Steps Using the Quadratic Formula

Combine -6u^{2} and 4u^{2} to get -2u^{2}.

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

Combine 5u and 7u to get 12u.

$$factor(-2u^{2}+12u-3+1)$$

Add -3 and 1 to get -2.

$$factor(-2u^{2}+12u-2)$$

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.

$$-2u^{2}+12u-2=0$$

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.

$$u=\frac{-12±\sqrt{12^{2}-4\left(-2\right)\left(-2\right)}}{2\left(-2\right)}$$

Square 12.

$$u=\frac{-12±\sqrt{144-4\left(-2\right)\left(-2\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$u=\frac{-12±\sqrt{144+8\left(-2\right)}}{2\left(-2\right)}$$

Multiply 8 times -2.

$$u=\frac{-12±\sqrt{144-16}}{2\left(-2\right)}$$

Add 144 to -16.

$$u=\frac{-12±\sqrt{128}}{2\left(-2\right)}$$

Take the square root of 128.

$$u=\frac{-12±8\sqrt{2}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$u=\frac{-12±8\sqrt{2}}{-4}$$

Now solve the equation u=\frac{-12±8\sqrt{2}}{-4} when ± is plus. Add -12 to 8\sqrt{2}.

$$u=\frac{8\sqrt{2}-12}{-4}$$

Divide -12+8\sqrt{2} by -4.

$$u=3-2\sqrt{2}$$

Now solve the equation u=\frac{-12±8\sqrt{2}}{-4} when ± is minus. Subtract 8\sqrt{2} from -12.

$$u=\frac{-8\sqrt{2}-12}{-4}$$

Divide -12-8\sqrt{2} by -4.

$$u=2\sqrt{2}+3$$

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

$$-2u^{2}+12u-2=-2\left(u-\left(3-2\sqrt{2}\right)\right)\left(u-\left(2\sqrt{2}+3\right)\right)$$