Solve {r}{x^2-36x}{+288}

$$\left. \begin{array} { r } { x ^ { 2 } - 36 x } \\ { + 288 } \end{array} \right.$$

Factor

$$\left(x-24\right)\left(x-12\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+288. To find a and b, set up a system to be solved.

$$a+b=-36 ab=1\times 288=288$$

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 288.

$$-1,-288 -2,-144 -3,-96 -4,-72 -6,-48 -8,-36 -9,-32 -12,-24 -16,-18$$

Calculate the sum for each pair.

$$-1-288=-289 -2-144=-146 -3-96=-99 -4-72=-76 -6-48=-54 -8-36=-44 -9-32=-41 -12-24=-36 -16-18=-34$$

The solution is the pair that gives sum -36.

$$a=-24 b=-12$$

Rewrite x^{2}-36x+288 as \left(x^{2}-24x\right)+\left(-12x+288\right).

$$\left(x^{2}-24x\right)+\left(-12x+288\right)$$

Factor out x in the first and -12 in the second group.

$$x\left(x-24\right)-12\left(x-24\right)$$

Factor out common term x-24 by using distributive property.

$$\left(x-24\right)\left(x-12\right)$$

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.

$$x^{2}-36x+288=0$$

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.

$$x=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 288}}{2}$$

Square -36.

$$x=\frac{-\left(-36\right)±\sqrt{1296-4\times 288}}{2}$$

Multiply -4 times 288.

$$x=\frac{-\left(-36\right)±\sqrt{1296-1152}}{2}$$

Add 1296 to -1152.

$$x=\frac{-\left(-36\right)±\sqrt{144}}{2}$$

Take the square root of 144.

$$x=\frac{-\left(-36\right)±12}{2}$$

The opposite of -36 is 36.

$$x=\frac{36±12}{2}$$

Now solve the equation x=\frac{36±12}{2} when ± is plus. Add 36 to 12.

$$x=\frac{48}{2}$$

Divide 48 by 2.

$$x=24$$

Now solve the equation x=\frac{36±12}{2} when ± is minus. Subtract 12 from 36.

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

Divide 24 by 2.

$$x=12$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 24 for x_{1} and 12 for x_{2}.

$$x^{2}-36x+288=\left(x-24\right)\left(x-12\right)$$

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.

$$x ^ 2 -36x +288 = 0$$

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

$$r + s = 36 rs = 288$$

Two numbers r and s sum up to 36 exactly when the average of the two numbers is \frac{1}{2}*36 = 18. 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>

$$r = 18 - u s = 18 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 288

$$(18 - u) (18 + u) = 288$$

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

$$324 - u^2 = 288$$

Simplify the expression by subtracting 324 on both sides

$$-u^2 = 288-324 = -36$$

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

$$u^2 = 36 u = \pm\sqrt{36} = \pm 6$$

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

$$r =18 - 6 = 12 s = 18 + 6 = 24$$

Evaluate

$$\left(x-24\right)\left(x-12\right)$$