Solve m^2+10m-24?

$$m ^ { 2 } + 10 m - 24 ?$$

Factor

$$\left(m-2\right)\left(m+12\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as m^{2}+am+bm-24. To find a and b, set up a system to be solved.

$$a+b=10 ab=1\left(-24\right)=-24$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -24.

$$-1,24 -2,12 -3,8 -4,6$$

Calculate the sum for each pair.

$$-1+24=23 -2+12=10 -3+8=5 -4+6=2$$

The solution is the pair that gives sum 10.

$$a=-2 b=12$$

Rewrite m^{2}+10m-24 as \left(m^{2}-2m\right)+\left(12m-24\right).

$$\left(m^{2}-2m\right)+\left(12m-24\right)$$

Factor out m in the first and 12 in the second group.

$$m\left(m-2\right)+12\left(m-2\right)$$

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

$$\left(m-2\right)\left(m+12\right)$$

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.

$$m^{2}+10m-24=0$$

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.

$$m=\frac{-10±\sqrt{10^{2}-4\left(-24\right)}}{2}$$

Square 10.

$$m=\frac{-10±\sqrt{100-4\left(-24\right)}}{2}$$

Multiply -4 times -24.

$$m=\frac{-10±\sqrt{100+96}}{2}$$

Add 100 to 96.

$$m=\frac{-10±\sqrt{196}}{2}$$

Take the square root of 196.

$$m=\frac{-10±14}{2}$$

Now solve the equation m=\frac{-10±14}{2} when ± is plus. Add -10 to 14.

$$m=\frac{4}{2}$$

Divide 4 by 2.

$$m=2$$

Now solve the equation m=\frac{-10±14}{2} when ± is minus. Subtract 14 from -10.

$$m=-\frac{24}{2}$$

Divide -24 by 2.

$$m=-12$$

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 -12 for x_{2}.

$$m^{2}+10m-24=\left(m-2\right)\left(m-\left(-12\right)\right)$$

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

$$m^{2}+10m-24=\left(m-2\right)\left(m+12\right)$$

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.

$$x ^ 2 +10x -24 = 0$$

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

$$r + s = -10 rs = -24$$

Two numbers r and s sum up to -10 exactly when the average of the two numbers is \frac{1}{2}*-10 = -5. 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>

$$r = -5 - u s = -5 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -24

$$(-5 - u) (-5 + u) = -24$$

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

$$25 - u^2 = -24$$

Simplify the expression by subtracting 25 on both sides

$$-u^2 = -24-25 = -49$$

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

$$u^2 = 49 u = \pm\sqrt{49} = \pm 7$$

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

$$r =-5 - 7 = -12 s = -5 + 7 = 2$$

Evaluate

$$\left(m-2\right)\left(m+12\right)$$