Factoring with an X

This is the method I was taught to factor with, and I still use it to this day. I can't find the proper name of this¹⁾ so X-Factoring is what I'll call it.

This tutorial assumes the standard form quadratic $ax^2 + bx + c$. For the steps below, I'll use the example quadratic $8x^2 - 72x + 162$.

Here's a diagram if the X if you get confused.

The first step before even starting to factor is to look for a GCF in all three terms, it will simplify the problem immensely if there is an $a$ value²⁾. In the case of our example, $8x^2 - 72x + 162$ can be simplified down to $2(4x^2 - 36x + 81)$. This may save you from factoring all together³⁾.

The first real step in factoring is to draw an X. In the top section, place your $c$ value in, and for the bottom, your $b$ value. For our example (after GCF factoring), the $c$ value of 81 goes on top, and the $b$ value of 36 goes below.

Now, for the two multipliers. If there is no $a$ value, this factoring method is extremely simple as it's practically plug 'n chug. However, if there is one, and especially if it is a composite number, it can get complicated quickly. For this example I chose one with an $a$ value to entice the brain a bit. If it is a prime number, the modifiers are easy as it has to be $1$ and the $a$ value. Note that the order of the modifiers does not matter⁴⁾. For composite numbers, you may have to guess and check for possible modifier/$x$/$y$ values, and here a factor tree for both the $a$ and $c$ values are extremely helpful. From anecdotal experience, try using the factors of $a$ that are closest together first (2 and 2 first before 4 and 1) as they tend to be correct more often than not. 2 and 2 are the correct multipliers for the example.

Now this is the bulk of the mental gymnastics, you must find two values, that when multiplied equal $c$, and add to equal $b$ after being multiplied by the multiplier below the number. Again, a factor tree is helpful if $c$ is a composite number. Likewise, if it its a prime number, the two values must be $1$ and the $c$ value. There isn't really a systematic method, you just have to quickly guess and check values until you find two that fit both requirements. You get better at this as you use this method more. For the example values of 9 for both $x$ and $y$ satisfy the requirements.

To double check, ensure that $xy = c$ and $y_{\text{modifier}} \times x + x_{\text{modifier}} \times y = b$. Or, in simpler terms, make sure $x$ and $y$ multiply to $c$ and add to $b$, accounting for multipliers below the numbers if present.

• $\boxed{9 \times 9 = 81} \checkmark$
• $\boxed{2 \times 9 + 2 \times 9 = 36} \checkmark$

To write the solution, put the x, y, and modifier values in the form $(x_{\text{modifier}}t + x)(y_{\text{modifier}}t + y)$⁵⁾. In other words, the x and y value use the multiplier on the opposite side of them, hence why they are labeled as such in the Tutorial Values section. The answer for the GCF'd example is $(2x + 9)(2x + 9)$⁶⁾. Thus, after including the GCF, $8x^2 - 72x + 162$ factors to $\boxed{2(2x + 9)(2x + 9)}$ which is our answer.

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