You know, sometimes math problems feel like a locked door, and factoring is like finding the right key. Today, we're going to tackle one such puzzle: how to factor the expression x² - 11x + 30.
It might look a bit daunting at first glance, but let's break it down. When we're asked to factor a quadratic expression like this – one with an x² term, an x term, and a constant – we're essentially trying to rewrite it as a product of two simpler expressions, usually two binomials.
Think of it like this: we're looking for two numbers that, when multiplied together, give us the constant term (which is 30 in this case), and when added together, give us the coefficient of the x term (which is -11).
So, our mission is to find two numbers, let's call them 'a' and 'b', such that:
a * b = 30 a + b = -11
Let's start by looking at the factors of 30. We have pairs like (1, 30), (2, 15), (3, 10), and (5, 6). Since our target sum is negative (-11) and our product is positive (30), this tells us that both numbers must be negative. Why? Because a negative times a negative is a positive, and a negative plus a negative is also a negative.
So, let's revisit our factor pairs, but this time with negative signs:
(-1) * (-30) = 30. (-1) + (-30) = -31. Nope, not quite. (-2) * (-15) = 30. (-2) + (-15) = -17. Still not there. (-3) * (-10) = 30. (-3) + (-10) = -13. Getting closer! (-5) * (-6) = 30. (-5) + (-6) = -11. Aha! We found them!
The two numbers we're looking for are -5 and -6.
Now that we have our magic numbers, we can use them to write our factored form. Since we found -5 and -6, our factored expression will be (x - 5)(x - 6).
To be absolutely sure, we can always expand this back out to check our work. Using the FOIL method (First, Outer, Inner, Last):
First: x * x = x² Outer: x * (-6) = -6x Inner: (-5) * x = -5x Last: (-5) * (-6) = 30
Putting it all together: x² - 6x - 5x + 30. Combining the like terms (-6x and -5x), we get x² - 11x + 30. And there you have it – it matches our original expression perfectly!
So, the factored form of x² - 11x + 30 is indeed (x - 5)(x - 6). It's a satisfying feeling when you can take something complex and break it down into its simpler components, isn't it?
