Solve x^2-23x+132

$$x ^ { 2 } - 23 x + 132$$

Factor

$$\left(x-12\right)\left(x-11\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+132. To find a and b, set up a system to be solved.

$$a+b=-23 ab=1\times 132=132$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 132.

$$-1,-132 -2,-66 -3,-44 -4,-33 -6,-22 -11,-12$$

Calculate the sum for each pair.

$$-1-132=-133 -2-66=-68 -3-44=-47 -4-33=-37 -6-22=-28 -11-12=-23$$

The solution is the pair that gives sum -23.

$$a=-12 b=-11$$

Rewrite x^{2}-23x+132 as \left(x^{2}-12x\right)+\left(-11x+132\right).

$$\left(x^{2}-12x\right)+\left(-11x+132\right)$$

Factor out x in the first and -11 in the second group.

$$x\left(x-12\right)-11\left(x-12\right)$$

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

$$\left(x-12\right)\left(x-11\right)$$

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.

$$x^{2}-23x+132=0$$

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.

$$x=\frac{-\left(-23\right)±\sqrt{\left(-23\right)^{2}-4\times 132}}{2}$$

Square -23.

$$x=\frac{-\left(-23\right)±\sqrt{529-4\times 132}}{2}$$

Multiply -4 times 132.

$$x=\frac{-\left(-23\right)±\sqrt{529-528}}{2}$$

Add 529 to -528.

$$x=\frac{-\left(-23\right)±\sqrt{1}}{2}$$

Take the square root of 1.

$$x=\frac{-\left(-23\right)±1}{2}$$

The opposite of -23 is 23.

$$x=\frac{23±1}{2}$$

Now solve the equation x=\frac{23±1}{2} when ± is plus. Add 23 to 1.

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

Divide 24 by 2.

$$x=12$$

Now solve the equation x=\frac{23±1}{2} when ± is minus. Subtract 1 from 23.

$$x=\frac{22}{2}$$

Divide 22 by 2.

$$x=11$$

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 11 for x_{2}.

$$x^{2}-23x+132=\left(x-12\right)\left(x-11\right)$$

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.

$$x ^ 2 -23x +132 = 0$$

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

$$r + s = 23 rs = 132$$

Two numbers r and s sum up to 23 exactly when the average of the two numbers is \frac{1}{2}*23 = \frac{23}{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>

$$r = \frac{23}{2} - u s = \frac{23}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 132

$$(\frac{23}{2} - u) (\frac{23}{2} + u) = 132$$

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

$$\frac{529}{4} - u^2 = 132$$

Simplify the expression by subtracting \frac{529}{4} on both sides

$$-u^2 = 132-\frac{529}{4} = -\frac{1}{4}$$

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

$$u^2 = \frac{1}{4} u = \pm\sqrt{\frac{1}{4}} = \pm \frac{1}{2}$$

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

$$r =\frac{23}{2} - \frac{1}{2} = 11 s = \frac{23}{2} + \frac{1}{2} = 12$$

Evaluate

$$\left(x-12\right)\left(x-11\right)$$