Solve 7v+v^2+12

$$7 v + v ^ { 2 } + 12$$

Factor

$$\left(v+3\right)\left(v+4\right)$$

Solution Steps

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$v^{2}+7v+12$$

Factor the expression by grouping. First, the expression needs to be rewritten as v^{2}+av+bv+12. To find a and b, set up a system to be solved.

$$a+b=7 ab=1\times 12=12$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 12.

$$1,12 2,6 3,4$$

Calculate the sum for each pair.

$$1+12=13 2+6=8 3+4=7$$

The solution is the pair that gives sum 7.

$$a=3 b=4$$

Rewrite v^{2}+7v+12 as \left(v^{2}+3v\right)+\left(4v+12\right).

$$\left(v^{2}+3v\right)+\left(4v+12\right)$$

Factor out v in the first and 4 in the second group.

$$v\left(v+3\right)+4\left(v+3\right)$$

Factor out common term v+3 by using distributive property.

$$\left(v+3\right)\left(v+4\right)$$

Steps Using the Quadratic Formula

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.

$$v^{2}+7v+12=0$$

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.

$$v=\frac{-7±\sqrt{7^{2}-4\times 12}}{2}$$

Square 7.

$$v=\frac{-7±\sqrt{49-4\times 12}}{2}$$

Multiply -4 times 12.

$$v=\frac{-7±\sqrt{49-48}}{2}$$

Add 49 to -48.

$$v=\frac{-7±\sqrt{1}}{2}$$

Take the square root of 1.

$$v=\frac{-7±1}{2}$$

Now solve the equation v=\frac{-7±1}{2} when ± is plus. Add -7 to 1.

$$v=-\frac{6}{2}$$

Divide -6 by 2.

$$v=-3$$

Now solve the equation v=\frac{-7±1}{2} when ± is minus. Subtract 1 from -7.

$$v=-\frac{8}{2}$$

Divide -8 by 2.

$$v=-4$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -3 for x_{1} and -4 for x_{2}.

$$v^{2}+7v+12=\left(v-\left(-3\right)\right)\left(v-\left(-4\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$v^{2}+7v+12=\left(v+3\right)\left(v+4\right)$$

Evaluate

$$\left(v+3\right)\left(v+4\right)$$