Solve 3m^2+14m+16

$$3 m ^ { 2 } + 14 m + 16$$

Factor

$$\left(m+2\right)\left(3m+8\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 3m^{2}+am+bm+16. To find a and b, set up a system to be solved.

$$a+b=14 ab=3\times 16=48$$

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 48.

$$1,48 2,24 3,16 4,12 6,8$$

Calculate the sum for each pair.

$$1+48=49 2+24=26 3+16=19 4+12=16 6+8=14$$

The solution is the pair that gives sum 14.

$$a=6 b=8$$

Rewrite 3m^{2}+14m+16 as \left(3m^{2}+6m\right)+\left(8m+16\right).

$$\left(3m^{2}+6m\right)+\left(8m+16\right)$$

Factor out 3m in the first and 8 in the second group.

$$3m\left(m+2\right)+8\left(m+2\right)$$

Factor out common term m+2 by using distributive property.

$$\left(m+2\right)\left(3m+8\right)$$

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.

$$3m^{2}+14m+16=0$$

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.

$$m=\frac{-14±\sqrt{14^{2}-4\times 3\times 16}}{2\times 3}$$

Square 14.

$$m=\frac{-14±\sqrt{196-4\times 3\times 16}}{2\times 3}$$

Multiply -4 times 3.

$$m=\frac{-14±\sqrt{196-12\times 16}}{2\times 3}$$

Multiply -12 times 16.

$$m=\frac{-14±\sqrt{196-192}}{2\times 3}$$

Add 196 to -192.

$$m=\frac{-14±\sqrt{4}}{2\times 3}$$

Take the square root of 4.

$$m=\frac{-14±2}{2\times 3}$$

Multiply 2 times 3.

$$m=\frac{-14±2}{6}$$

Now solve the equation m=\frac{-14±2}{6} when ± is plus. Add -14 to 2.

$$m=-\frac{12}{6}$$

Divide -12 by 6.

$$m=-2$$

Now solve the equation m=\frac{-14±2}{6} when ± is minus. Subtract 2 from -14.

$$m=-\frac{16}{6}$$

Reduce the fraction \frac{-16}{6} to lowest terms by extracting and canceling out 2.

$$m=-\frac{8}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -2 for x_{1} and -\frac{8}{3} for x_{2}.

$$3m^{2}+14m+16=3\left(m-\left(-2\right)\right)\left(m-\left(-\frac{8}{3}\right)\right)$$

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

$$3m^{2}+14m+16=3\left(m+2\right)\left(m+\frac{8}{3}\right)$$

Add \frac{8}{3} to m by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$3m^{2}+14m+16=3\left(m+2\right)\times \frac{3m+8}{3}$$

Cancel out 3, the greatest common factor in 3 and 3.

$$3m^{2}+14m+16=\left(m+2\right)\left(3m+8\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 3

$$x ^ 2 +\frac{14}{3}x +\frac{16}{3} = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = -\frac{14}{3} rs = \frac{16}{3}$$

Two numbers r and s sum up to -\frac{14}{3} exactly when the average of the two numbers is \frac{1}{2}*-\frac{14}{3} = -\frac{7}{3}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -\frac{7}{3} - u s = -\frac{7}{3} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{16}{3}

$$(-\frac{7}{3} - u) (-\frac{7}{3} + u) = \frac{16}{3}$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{49}{9} - u^2 = \frac{16}{3}$$

Simplify the expression by subtracting \frac{49}{9} on both sides

$$-u^2 = \frac{16}{3}-\frac{49}{9} = -\frac{1}{9}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{1}{9} u = \pm\sqrt{\frac{1}{9}} = \pm \frac{1}{3}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =-\frac{7}{3} - \frac{1}{3} = -2.667 s = -\frac{7}{3} + \frac{1}{3} = -2.000$$

Evaluate

$$\left(m+2\right)\left(3m+8\right)$$