Solve 3+5v^2+30v+6

$$3 + 5 v ^ { 2 } + 30 v + 6$$

Factor

$$5\left(v-\left(-\frac{6\sqrt{5}}{5}-3\right)\right)\left(v-\left(\frac{6\sqrt{5}}{5}-3\right)\right)$$

Steps Using the Quadratic Formula

Add 3 and 6 to get 9.

$$factor(9+5v^{2}+30v)$$

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.

$$5v^{2}+30v+9=0$$

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.

$$v=\frac{-30±\sqrt{30^{2}-4\times 5\times 9}}{2\times 5}$$

Square 30.

$$v=\frac{-30±\sqrt{900-4\times 5\times 9}}{2\times 5}$$

Multiply -4 times 5.

$$v=\frac{-30±\sqrt{900-20\times 9}}{2\times 5}$$

Multiply -20 times 9.

$$v=\frac{-30±\sqrt{900-180}}{2\times 5}$$

Add 900 to -180.

$$v=\frac{-30±\sqrt{720}}{2\times 5}$$

Take the square root of 720.

$$v=\frac{-30±12\sqrt{5}}{2\times 5}$$

Multiply 2 times 5.

$$v=\frac{-30±12\sqrt{5}}{10}$$

Now solve the equation v=\frac{-30±12\sqrt{5}}{10} when ± is plus. Add -30 to 12\sqrt{5}.

$$v=\frac{12\sqrt{5}-30}{10}$$

Divide -30+12\sqrt{5} by 10.

$$v=\frac{6\sqrt{5}}{5}-3$$

Now solve the equation v=\frac{-30±12\sqrt{5}}{10} when ± is minus. Subtract 12\sqrt{5} from -30.

$$v=\frac{-12\sqrt{5}-30}{10}$$

Divide -30-12\sqrt{5} by 10.

$$v=-\frac{6\sqrt{5}}{5}-3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -3+\frac{6\sqrt{5}}{5} for x_{1} and -3-\frac{6\sqrt{5}}{5} for x_{2}.

$$5v^{2}+30v+9=5\left(v-\left(\frac{6\sqrt{5}}{5}-3\right)\right)\left(v-\left(-\frac{6\sqrt{5}}{5}-3\right)\right)$$

Evaluate

$$5v^{2}+30v+9$$

Solution Steps

Add 3 and 6 to get 9.

$$9+5v^{2}+30v$$