Solve 6m^2+48m+72

$$6 m ^ { 2 } + 48 m + 72$$

Factor

$$6\left(m+2\right)\left(m+6\right)$$

Solution Steps

Factor out 6.

$$6\left(m^{2}+8m+12\right)$$

Consider m^{2}+8m+12. Factor the expression by grouping. First, the expression needs to be rewritten as m^{2}+am+bm+12. To find a and b, set up a system to be solved.

$$a+b=8 ab=1\times 12=12$$

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 8.

$$a=2 b=6$$

Rewrite m^{2}+8m+12 as \left(m^{2}+2m\right)+\left(6m+12\right).

$$\left(m^{2}+2m\right)+\left(6m+12\right)$$

Factor out m in the first and 6 in the second group.

$$m\left(m+2\right)+6\left(m+2\right)$$

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

$$\left(m+2\right)\left(m+6\right)$$

Rewrite the complete factored expression.

$$6\left(m+2\right)\left(m+6\right)$$

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.

$$6m^{2}+48m+72=0$$

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.

$$m=\frac{-48±\sqrt{48^{2}-4\times 6\times 72}}{2\times 6}$$

Square 48.

$$m=\frac{-48±\sqrt{2304-4\times 6\times 72}}{2\times 6}$$

Multiply -4 times 6.

$$m=\frac{-48±\sqrt{2304-24\times 72}}{2\times 6}$$

Multiply -24 times 72.

$$m=\frac{-48±\sqrt{2304-1728}}{2\times 6}$$

Add 2304 to -1728.

$$m=\frac{-48±\sqrt{576}}{2\times 6}$$

Take the square root of 576.

$$m=\frac{-48±24}{2\times 6}$$

Multiply 2 times 6.

$$m=\frac{-48±24}{12}$$

Now solve the equation m=\frac{-48±24}{12} when ± is plus. Add -48 to 24.

$$m=-\frac{24}{12}$$

Divide -24 by 12.

$$m=-2$$

Now solve the equation m=\frac{-48±24}{12} when ± is minus. Subtract 24 from -48.

$$m=-\frac{72}{12}$$

Divide -72 by 12.

$$m=-6$$

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 -6 for x_{2}.

$$6m^{2}+48m+72=6\left(m-\left(-2\right)\right)\left(m-\left(-6\right)\right)$$

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

$$6m^{2}+48m+72=6\left(m+2\right)\left(m+6\right)$$

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 6

$$x ^ 2 +8x +12 = 0$$

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

$$r + s = -8 rs = 12$$

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

$$r = -4 - u s = -4 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 12

$$(-4 - u) (-4 + u) = 12$$

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

$$16 - u^2 = 12$$

Simplify the expression by subtracting 16 on both sides

$$-u^2 = 12-16 = -4$$

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

$$u^2 = 4 u = \pm\sqrt{4} = \pm 2$$

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

$$r =-4 - 2 = -6 s = -4 + 2 = -2$$

Evaluate

$$6\left(m+2\right)\left(m+6\right)$$