Solve (-5v^2-v+3)+(-4v^2+6)

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

Evaluate

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

Solution Steps

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

$$-9v^{2}-v+3+6$$

Add 3 and 6 to get 9.

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

Factor

$$-9\left(v-\frac{-5\sqrt{13}-1}{18}\right)\left(v-\frac{5\sqrt{13}-1}{18}\right)$$

Steps Using the Quadratic Formula

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

$$factor(-9v^{2}-v+3+6)$$

Add 3 and 6 to get 9.

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

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.

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

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.

$$v=\frac{-\left(-1\right)±\sqrt{1-4\left(-9\right)\times 9}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$v=\frac{-\left(-1\right)±\sqrt{1+36\times 9}}{2\left(-9\right)}$$

Multiply 36 times 9.

$$v=\frac{-\left(-1\right)±\sqrt{1+324}}{2\left(-9\right)}$$

Add 1 to 324.

$$v=\frac{-\left(-1\right)±\sqrt{325}}{2\left(-9\right)}$$

Take the square root of 325.

$$v=\frac{-\left(-1\right)±5\sqrt{13}}{2\left(-9\right)}$$

The opposite of -1 is 1.

$$v=\frac{1±5\sqrt{13}}{2\left(-9\right)}$$

Multiply 2 times -9.

$$v=\frac{1±5\sqrt{13}}{-18}$$

Now solve the equation v=\frac{1±5\sqrt{13}}{-18} when ± is plus. Add 1 to 5\sqrt{13}.

$$v=\frac{5\sqrt{13}+1}{-18}$$

Divide 1+5\sqrt{13} by -18.

$$v=\frac{-5\sqrt{13}-1}{18}$$

Now solve the equation v=\frac{1±5\sqrt{13}}{-18} when ± is minus. Subtract 5\sqrt{13} from 1.

$$v=\frac{1-5\sqrt{13}}{-18}$$

Divide 1-5\sqrt{13} by -18.

$$v=\frac{5\sqrt{13}-1}{18}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-1-5\sqrt{13}}{18} for x_{1} and \frac{-1+5\sqrt{13}}{18} for x_{2}.

$$-9v^{2}-v+9=-9\left(v-\frac{-5\sqrt{13}-1}{18}\right)\left(v-\frac{5\sqrt{13}-1}{18}\right)$$