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

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

Evaluate

$$7-4u-13u^{2}$$

Solution Steps

Combine -7u^{2} and -6u^{2} to get -13u^{2}.

$$-13u^{2}-4u+5+2$$

Add 5 and 2 to get 7.

$$-13u^{2}-4u+7$$

Factor

$$-13\left(u-\frac{-\sqrt{95}-2}{13}\right)\left(u-\frac{\sqrt{95}-2}{13}\right)$$

Steps Using the Quadratic Formula

Combine -7u^{2} and -6u^{2} to get -13u^{2}.

$$factor(-13u^{2}-4u+5+2)$$

Add 5 and 2 to get 7.

$$factor(-13u^{2}-4u+7)$$

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.

$$-13u^{2}-4u+7=0$$

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.

$$u=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-13\right)\times 7}}{2\left(-13\right)}$$

Square -4.

$$u=\frac{-\left(-4\right)±\sqrt{16-4\left(-13\right)\times 7}}{2\left(-13\right)}$$

Multiply -4 times -13.

$$u=\frac{-\left(-4\right)±\sqrt{16+52\times 7}}{2\left(-13\right)}$$

Multiply 52 times 7.

$$u=\frac{-\left(-4\right)±\sqrt{16+364}}{2\left(-13\right)}$$

Add 16 to 364.

$$u=\frac{-\left(-4\right)±\sqrt{380}}{2\left(-13\right)}$$

Take the square root of 380.

$$u=\frac{-\left(-4\right)±2\sqrt{95}}{2\left(-13\right)}$$

The opposite of -4 is 4.

$$u=\frac{4±2\sqrt{95}}{2\left(-13\right)}$$

Multiply 2 times -13.

$$u=\frac{4±2\sqrt{95}}{-26}$$

Now solve the equation u=\frac{4±2\sqrt{95}}{-26} when ± is plus. Add 4 to 2\sqrt{95}.

$$u=\frac{2\sqrt{95}+4}{-26}$$

Divide 4+2\sqrt{95} by -26.

$$u=\frac{-\sqrt{95}-2}{13}$$

Now solve the equation u=\frac{4±2\sqrt{95}}{-26} when ± is minus. Subtract 2\sqrt{95} from 4.

$$u=\frac{4-2\sqrt{95}}{-26}$$

Divide 4-2\sqrt{95} by -26.

$$u=\frac{\sqrt{95}-2}{13}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-2-\sqrt{95}}{13} for x_{1} and \frac{-2+\sqrt{95}}{13} for x_{2}.

$$-13u^{2}-4u+7=-13\left(u-\frac{-\sqrt{95}-2}{13}\right)\left(u-\frac{\sqrt{95}-2}{13}\right)$$