Solve n^2-14n-72

$$n ^ { 2 } - 14 n - 72$$

Factor

$$\left(n-18\right)\left(n+4\right)$$

Solution Steps

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

$$a+b=-14 ab=1\left(-72\right)=-72$$

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 -14.

$$a=-18 b=4$$

Rewrite n^{2}-14n-72 as \left(n^{2}-18n\right)+\left(4n-72\right).

$$\left(n^{2}-18n\right)+\left(4n-72\right)$$

Factor out n in the first and 4 in the second group.

$$n\left(n-18\right)+4\left(n-18\right)$$

Factor out common term n-18 by using distributive property.

$$\left(n-18\right)\left(n+4\right)$$

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.

$$n^{2}-14n-72=0$$

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.

$$n=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\left(-72\right)}}{2}$$

Square -14.

$$n=\frac{-\left(-14\right)±\sqrt{196-4\left(-72\right)}}{2}$$

Multiply -4 times -72.

$$n=\frac{-\left(-14\right)±\sqrt{196+288}}{2}$$

Add 196 to 288.

$$n=\frac{-\left(-14\right)±\sqrt{484}}{2}$$

Take the square root of 484.

$$n=\frac{-\left(-14\right)±22}{2}$$

The opposite of -14 is 14.

$$n=\frac{14±22}{2}$$

Now solve the equation n=\frac{14±22}{2} when ± is plus. Add 14 to 22.

$$n=\frac{36}{2}$$

Divide 36 by 2.

$$n=18$$

Now solve the equation n=\frac{14±22}{2} when ± is minus. Subtract 22 from 14.

$$n=-\frac{8}{2}$$

Divide -8 by 2.

$$n=-4$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 18 for x_{1} and -4 for x_{2}.

$$n^{2}-14n-72=\left(n-18\right)\left(n-\left(-4\right)\right)$$

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

$$n^{2}-14n-72=\left(n-18\right)\left(n+4\right)$$

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.

$$x ^ 2 -14x -72 = 0$$

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

$$r + s = 14 rs = -72$$

Two numbers r and s sum up to 14 exactly when the average of the two numbers is \frac{1}{2}*14 = 7. 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>

$$r = 7 - u s = 7 + u$$

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

$$(7 - u) (7 + u) = -72$$

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

$$49 - u^2 = -72$$

Simplify the expression by subtracting 49 on both sides

$$-u^2 = -72-49 = -121$$

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

$$u^2 = 121 u = \pm\sqrt{121} = \pm 11$$

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

$$r =7 - 11 = -4 s = 7 + 11 = 18$$

Evaluate

$$\left(n-18\right)\left(n+4\right)$$