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

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

Evaluate

$$10v^{2}-3v-2$$

Solution Steps

Combine 4v^{2} and 6v^{2} to get 10v^{2}.

$$10v^{2}+5-3v-7$$

Subtract 7 from 5 to get -2.

$$10v^{2}-2-3v$$

Factor

$$10\left(v-\frac{3-\sqrt{89}}{20}\right)\left(v-\frac{\sqrt{89}+3}{20}\right)$$

Steps Using the Quadratic Formula

Combine 4v^{2} and 6v^{2} to get 10v^{2}.

$$factor(10v^{2}+5-3v-7)$$

Subtract 7 from 5 to get -2.

$$factor(10v^{2}-2-3v)$$

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.

$$10v^{2}-3v-2=0$$

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.

$$v=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 10\left(-2\right)}}{2\times 10}$$

Square -3.

$$v=\frac{-\left(-3\right)±\sqrt{9-4\times 10\left(-2\right)}}{2\times 10}$$

Multiply -4 times 10.

$$v=\frac{-\left(-3\right)±\sqrt{9-40\left(-2\right)}}{2\times 10}$$

Multiply -40 times -2.

$$v=\frac{-\left(-3\right)±\sqrt{9+80}}{2\times 10}$$

Add 9 to 80.

$$v=\frac{-\left(-3\right)±\sqrt{89}}{2\times 10}$$

The opposite of -3 is 3.

$$v=\frac{3±\sqrt{89}}{2\times 10}$$

Multiply 2 times 10.

$$v=\frac{3±\sqrt{89}}{20}$$

Now solve the equation v=\frac{3±\sqrt{89}}{20} when ± is plus. Add 3 to \sqrt{89}.

$$v=\frac{\sqrt{89}+3}{20}$$

Now solve the equation v=\frac{3±\sqrt{89}}{20} when ± is minus. Subtract \sqrt{89} from 3.

$$v=\frac{3-\sqrt{89}}{20}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3+\sqrt{89}}{20} for x_{1} and \frac{3-\sqrt{89}}{20} for x_{2}.

$$10v^{2}-3v-2=10\left(v-\frac{\sqrt{89}+3}{20}\right)\left(v-\frac{3-\sqrt{89}}{20}\right)$$