Solve x^2-32x-144

$$x ^ { 2 } - 32 x - 144$$

Factor

$$\left(x-36\right)\left(x+4\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-144. To find a and b, set up a system to be solved.

$$a+b=-32 ab=1\left(-144\right)=-144$$

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 -144.

$$1,-144 2,-72 3,-48 4,-36 6,-24 8,-18 9,-16 12,-12$$

Calculate the sum for each pair.

$$1-144=-143 2-72=-70 3-48=-45 4-36=-32 6-24=-18 8-18=-10 9-16=-7 12-12=0$$

The solution is the pair that gives sum -32.

$$a=-36 b=4$$

Rewrite x^{2}-32x-144 as \left(x^{2}-36x\right)+\left(4x-144\right).

$$\left(x^{2}-36x\right)+\left(4x-144\right)$$

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

$$x\left(x-36\right)+4\left(x-36\right)$$

Factor out common term x-36 by using distributive property.

$$\left(x-36\right)\left(x+4\right)$$

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.

$$x^{2}-32x-144=0$$

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.

$$x=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\left(-144\right)}}{2}$$

Square -32.

$$x=\frac{-\left(-32\right)±\sqrt{1024-4\left(-144\right)}}{2}$$

Multiply -4 times -144.

$$x=\frac{-\left(-32\right)±\sqrt{1024+576}}{2}$$

Add 1024 to 576.

$$x=\frac{-\left(-32\right)±\sqrt{1600}}{2}$$

Take the square root of 1600.

$$x=\frac{-\left(-32\right)±40}{2}$$

The opposite of -32 is 32.

$$x=\frac{32±40}{2}$$

Now solve the equation x=\frac{32±40}{2} when ± is plus. Add 32 to 40.

$$x=\frac{72}{2}$$

Divide 72 by 2.

$$x=36$$

Now solve the equation x=\frac{32±40}{2} when ± is minus. Subtract 40 from 32.

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

Divide -8 by 2.

$$x=-4$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 36 for x_{1} and -4 for x_{2}.

$$x^{2}-32x-144=\left(x-36\right)\left(x-\left(-4\right)\right)$$

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

$$x^{2}-32x-144=\left(x-36\right)\left(x+4\right)$$

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.

$$x ^ 2 -32x -144 = 0$$

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

$$r + s = 32 rs = -144$$

Two numbers r and s sum up to 32 exactly when the average of the two numbers is \frac{1}{2}*32 = 16. 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>

$$r = 16 - u s = 16 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -144

$$(16 - u) (16 + u) = -144$$

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

$$256 - u^2 = -144$$

Simplify the expression by subtracting 256 on both sides

$$-u^2 = -144-256 = -400$$

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

$$u^2 = 400 u = \pm\sqrt{400} = \pm 20$$

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

$$r =16 - 20 = -4 s = 16 + 20 = 36$$

Evaluate

$$\left(x-36\right)\left(x+4\right)$$