You know, sometimes math problems can feel like a bit of a puzzle, can't they? You're presented with an expression, like x² + 3x - 10, and the goal is to "factorize" it. It sounds a bit technical, but at its heart, it's about breaking down something complex into its simpler building blocks – like finding the ingredients that make up a recipe.
So, what does it mean to factorize x² + 3x - 10? Think of it as the reverse of expanding. When you multiply two simple expressions together, like (x + a)(x + b), you get a quadratic expression. Factoring is the process of going back from that quadratic expression to the original two simpler ones.
For an expression in the form x² + bx + c, like ours where b is 3 and c is -10, we're looking for two numbers. These two special numbers need to do two things simultaneously: their product must equal the constant term (c, which is -10 in our case), and their sum must equal the coefficient of the x term (b, which is 3).
Let's brainstorm. We need a pair of numbers that multiply to -10. Possible pairs include (1, -10), (-1, 10), (2, -5), and (-2, 5). Now, let's check their sums:
- 1 + (-10) = -9
- -1 + 10 = 9
- 2 + (-5) = -3
- -2 + 5 = 3
Ah, there it is! The pair -2 and 5 fits the bill perfectly. Their product is -10, and their sum is 3.
Once we've found these two numbers, say p and q, the factored form of x² + bx + c is simply (x + p)(x + q). In our case, with p = -2 and q = 5, the factored expression becomes (x - 2)(x + 5).
It's always a good idea to double-check our work, right? Let's expand (x - 2)(x + 5) to see if we get back to our original expression:
(x - 2)(x + 5) = x * x + x * 5 - 2 * x - 2 * 5
= x² + 5x - 2x - 10
= x² + 3x - 10
And there we have it! It matches perfectly. So, the factorization of x² + 3x - 10 is indeed (x - 2)(x + 5).
This method, often called "splitting the middle term" or finding two numbers that multiply to c and add to b, is a fundamental technique in algebra. It's like having a key that unlocks the structure of quadratic expressions, making them easier to understand and manipulate.
