Solve (-5v^2-3v+2)+(-4v^2+7)

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

Evaluate

$$9-3v-9v^{2}$$

Solution Steps

Combine -5v^{2} and -4v^{2} to get -9v^{2}.

$$-9v^{2}-3v+2+7$$

Add 2 and 7 to get 9.

$$-9v^{2}-3v+9$$

Factor

$$-9\left(v-\frac{-\sqrt{37}-1}{6}\right)\left(v-\frac{\sqrt{37}-1}{6}\right)$$

Steps Using the Quadratic Formula

Combine -5v^{2} and -4v^{2} to get -9v^{2}.

$$factor(-9v^{2}-3v+2+7)$$

Add 2 and 7 to get 9.

$$factor(-9v^{2}-3v+9)$$

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.

$$-9v^{2}-3v+9=0$$

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.

$$v=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-9\right)\times 9}}{2\left(-9\right)}$$

Square -3.

$$v=\frac{-\left(-3\right)±\sqrt{9-4\left(-9\right)\times 9}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$v=\frac{-\left(-3\right)±\sqrt{9+36\times 9}}{2\left(-9\right)}$$

Multiply 36 times 9.

$$v=\frac{-\left(-3\right)±\sqrt{9+324}}{2\left(-9\right)}$$

Add 9 to 324.

$$v=\frac{-\left(-3\right)±\sqrt{333}}{2\left(-9\right)}$$

Take the square root of 333.

$$v=\frac{-\left(-3\right)±3\sqrt{37}}{2\left(-9\right)}$$

The opposite of -3 is 3.

$$v=\frac{3±3\sqrt{37}}{2\left(-9\right)}$$

Multiply 2 times -9.

$$v=\frac{3±3\sqrt{37}}{-18}$$

Now solve the equation v=\frac{3±3\sqrt{37}}{-18} when ± is plus. Add 3 to 3\sqrt{37}.

$$v=\frac{3\sqrt{37}+3}{-18}$$

Divide 3+3\sqrt{37} by -18.

$$v=\frac{-\sqrt{37}-1}{6}$$

Now solve the equation v=\frac{3±3\sqrt{37}}{-18} when ± is minus. Subtract 3\sqrt{37} from 3.

$$v=\frac{3-3\sqrt{37}}{-18}$$

Divide 3-3\sqrt{37} by -18.

$$v=\frac{\sqrt{37}-1}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-1-\sqrt{37}}{6} for x_{1} and \frac{-1+\sqrt{37}}{6} for x_{2}.

$$-9v^{2}-3v+9=-9\left(v-\frac{-\sqrt{37}-1}{6}\right)\left(v-\frac{\sqrt{37}-1}{6}\right)$$