Solve 12m^2+32m+16

$$12 m ^ { 2 } + 32 m + 16$$

Factor

$$4\left(m+2\right)\left(3m+2\right)$$

Solution Steps

Factor out 4.

$$4\left(3m^{2}+8m+4\right)$$

Consider 3m^{2}+8m+4. Factor the expression by grouping. First, the expression needs to be rewritten as 3m^{2}+am+bm+4. To find a and b, set up a system to be solved.

$$a+b=8 ab=3\times 4=12$$

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 8.

$$a=2 b=6$$

Rewrite 3m^{2}+8m+4 as \left(3m^{2}+2m\right)+\left(6m+4\right).

$$\left(3m^{2}+2m\right)+\left(6m+4\right)$$

Factor out m in the first and 2 in the second group.

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

Factor out common term 3m+2 by using distributive property.

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

Rewrite the complete factored expression.

$$4\left(3m+2\right)\left(m+2\right)$$

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.

$$12m^{2}+32m+16=0$$

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.

$$m=\frac{-32±\sqrt{32^{2}-4\times 12\times 16}}{2\times 12}$$

Square 32.

$$m=\frac{-32±\sqrt{1024-4\times 12\times 16}}{2\times 12}$$

Multiply -4 times 12.

$$m=\frac{-32±\sqrt{1024-48\times 16}}{2\times 12}$$

Multiply -48 times 16.

$$m=\frac{-32±\sqrt{1024-768}}{2\times 12}$$

Add 1024 to -768.

$$m=\frac{-32±\sqrt{256}}{2\times 12}$$

Take the square root of 256.

$$m=\frac{-32±16}{2\times 12}$$

Multiply 2 times 12.

$$m=\frac{-32±16}{24}$$

Now solve the equation m=\frac{-32±16}{24} when ± is plus. Add -32 to 16.

$$m=-\frac{16}{24}$$

Reduce the fraction \frac{-16}{24} to lowest terms by extracting and canceling out 8.

$$m=-\frac{2}{3}$$

Now solve the equation m=\frac{-32±16}{24} when ± is minus. Subtract 16 from -32.

$$m=-\frac{48}{24}$$

Divide -48 by 24.

$$m=-2$$

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

$$12m^{2}+32m+16=12\left(m-\left(-\frac{2}{3}\right)\right)\left(m-\left(-2\right)\right)$$

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

$$12m^{2}+32m+16=12\left(m+\frac{2}{3}\right)\left(m+2\right)$$

Add \frac{2}{3} to m by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$12m^{2}+32m+16=12\times \frac{3m+2}{3}\left(m+2\right)$$

Cancel out 3, the greatest common factor in 12 and 3.

$$12m^{2}+32m+16=4\left(3m+2\right)\left(m+2\right)$$

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 12

$$x ^ 2 +\frac{8}{3}x +\frac{4}{3} = 0$$

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

$$r + s = -\frac{8}{3} rs = \frac{4}{3}$$

Two numbers r and s sum up to -\frac{8}{3} exactly when the average of the two numbers is \frac{1}{2}*-\frac{8}{3} = -\frac{4}{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>

$$r = -\frac{4}{3} - u s = -\frac{4}{3} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{4}{3}

$$(-\frac{4}{3} - u) (-\frac{4}{3} + u) = \frac{4}{3}$$

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

$$\frac{16}{9} - u^2 = \frac{4}{3}$$

Simplify the expression by subtracting \frac{16}{9} on both sides

$$-u^2 = \frac{4}{3}-\frac{16}{9} = -\frac{4}{9}$$

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

$$u^2 = \frac{4}{9} u = \pm\sqrt{\frac{4}{9}} = \pm \frac{2}{3}$$

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

$$r =-\frac{4}{3} - \frac{2}{3} = -2 s = -\frac{4}{3} + \frac{2}{3} = -0.667$$

Evaluate

$$4\left(m+2\right)\left(3m+2\right)$$