Solve m^2+2m-168

$$m ^ { 2 } + 2 m - 168$$

Factor

$$\left(m-12\right)\left(m+14\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as m^{2}+am+bm-168. To find a and b, set up a system to be solved.

$$a+b=2 ab=1\left(-168\right)=-168$$

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 -168.

$$-1,168 -2,84 -3,56 -4,42 -6,28 -7,24 -8,21 -12,14$$

Calculate the sum for each pair.

$$-1+168=167 -2+84=82 -3+56=53 -4+42=38 -6+28=22 -7+24=17 -8+21=13 -12+14=2$$

The solution is the pair that gives sum 2.

$$a=-12 b=14$$

Rewrite m^{2}+2m-168 as \left(m^{2}-12m\right)+\left(14m-168\right).

$$\left(m^{2}-12m\right)+\left(14m-168\right)$$

Factor out m in the first and 14 in the second group.

$$m\left(m-12\right)+14\left(m-12\right)$$

Factor out common term m-12 by using distributive property.

$$\left(m-12\right)\left(m+14\right)$$

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.

$$m^{2}+2m-168=0$$

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.

$$m=\frac{-2±\sqrt{2^{2}-4\left(-168\right)}}{2}$$

Square 2.

$$m=\frac{-2±\sqrt{4-4\left(-168\right)}}{2}$$

Multiply -4 times -168.

$$m=\frac{-2±\sqrt{4+672}}{2}$$

Add 4 to 672.

$$m=\frac{-2±\sqrt{676}}{2}$$

Take the square root of 676.

$$m=\frac{-2±26}{2}$$

Now solve the equation m=\frac{-2±26}{2} when ± is plus. Add -2 to 26.

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

Divide 24 by 2.

$$m=12$$

Now solve the equation m=\frac{-2±26}{2} when ± is minus. Subtract 26 from -2.

$$m=-\frac{28}{2}$$

Divide -28 by 2.

$$m=-14$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 12 for x_{1} and -14 for x_{2}.

$$m^{2}+2m-168=\left(m-12\right)\left(m-\left(-14\right)\right)$$

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

$$m^{2}+2m-168=\left(m-12\right)\left(m+14\right)$$

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.

$$x ^ 2 +2x -168 = 0$$

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

$$r + s = -2 rs = -168$$

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

$$r = -1 - u s = -1 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -168

$$(-1 - u) (-1 + u) = -168$$

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

$$1 - u^2 = -168$$

Simplify the expression by subtracting 1 on both sides

$$-u^2 = -168-1 = -169$$

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

$$u^2 = 169 u = \pm\sqrt{169} = \pm 13$$

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

$$r =-1 - 13 = -14 s = -1 + 13 = 12$$

Evaluate

$$\left(m-12\right)\left(m+14\right)$$