Solve m^2-21m-72

$$m ^ { 2 } - 21 m - 72$$

Factor

$$\left(m-24\right)\left(m+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as m^{2}+am+bm-72. To find a and b, set up a system to be solved.

$$a+b=-21 ab=1\left(-72\right)=-72$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -72.

$$1,-72 2,-36 3,-24 4,-18 6,-12 8,-9$$

Calculate the sum for each pair.

$$1-72=-71 2-36=-34 3-24=-21 4-18=-14 6-12=-6 8-9=-1$$

The solution is the pair that gives sum -21.

$$a=-24 b=3$$

Rewrite m^{2}-21m-72 as \left(m^{2}-24m\right)+\left(3m-72\right).

$$\left(m^{2}-24m\right)+\left(3m-72\right)$$

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

$$m\left(m-24\right)+3\left(m-24\right)$$

Factor out common term m-24 by using distributive property.

$$\left(m-24\right)\left(m+3\right)$$

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.

$$m^{2}-21m-72=0$$

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.

$$m=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\left(-72\right)}}{2}$$

Square -21.

$$m=\frac{-\left(-21\right)±\sqrt{441-4\left(-72\right)}}{2}$$

Multiply -4 times -72.

$$m=\frac{-\left(-21\right)±\sqrt{441+288}}{2}$$

Add 441 to 288.

$$m=\frac{-\left(-21\right)±\sqrt{729}}{2}$$

Take the square root of 729.

$$m=\frac{-\left(-21\right)±27}{2}$$

The opposite of -21 is 21.

$$m=\frac{21±27}{2}$$

Now solve the equation m=\frac{21±27}{2} when ± is plus. Add 21 to 27.

$$m=\frac{48}{2}$$

Divide 48 by 2.

$$m=24$$

Now solve the equation m=\frac{21±27}{2} when ± is minus. Subtract 27 from 21.

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

Divide -6 by 2.

$$m=-3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 24 for x_{1} and -3 for x_{2}.

$$m^{2}-21m-72=\left(m-24\right)\left(m-\left(-3\right)\right)$$

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

$$m^{2}-21m-72=\left(m-24\right)\left(m+3\right)$$

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.

$$x ^ 2 -21x -72 = 0$$

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

$$r + s = 21 rs = -72$$

Two numbers r and s sum up to 21 exactly when the average of the two numbers is \frac{1}{2}*21 = \frac{21}{2}. 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>

$$r = \frac{21}{2} - u s = \frac{21}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -72

$$(\frac{21}{2} - u) (\frac{21}{2} + u) = -72$$

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

$$\frac{441}{4} - u^2 = -72$$

Simplify the expression by subtracting \frac{441}{4} on both sides

$$-u^2 = -72-\frac{441}{4} = -\frac{729}{4}$$

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

$$u^2 = \frac{729}{4} u = \pm\sqrt{\frac{729}{4}} = \pm \frac{27}{2}$$

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

$$r =\frac{21}{2} - \frac{27}{2} = -3 s = \frac{21}{2} + \frac{27}{2} = 24$$

Evaluate

$$\left(m-24\right)\left(m+3\right)$$