What is the factorise of 2x2 + 7x + 3? Unpacking the Process for Everyday Math
Ever stared at a math problem and felt a little lost, especially when it involves those pesky 'x' terms? You're not alone! Many of us learned about factoring in school, but with the passage of time, the details can get a bit fuzzy. Today, we're going to break down exactly what the factorisation of 2x2 + 7x + 3 is, step-by-step, so you can feel confident tackling similar problems.
Understanding the Goal: What Does "Factorise" Mean?
Before we dive into the specifics of 2x2 + 7x + 3, let's clarify what "factorising" means in mathematics. Think of it like un-multiplying. When you factor a number, you break it down into its smaller building blocks that, when multiplied together, give you the original number. For example, the factors of 12 are 2, 2, and 3 because 2 * 2 * 3 = 12.
When we factor an algebraic expression, like 2x2 + 7x + 3, we're doing the same thing but with terms that include variables (like 'x') and exponents. Our goal is to rewrite the expression as a product of two or more simpler expressions, typically binomials (expressions with two terms).
The Specific Expression: 2x2 + 7x + 3
The expression we're working with is a quadratic trinomial. This means it has three terms and the highest power of 'x' is 2 (represented by x2).
- 2x2: This is the first term. The '2' is the coefficient, and 'x2' is the variable part.
- +7x: This is the middle term, with a coefficient of 7 and the variable 'x'.
- +3: This is the constant term, with no variable attached.
Methods for Factorising Quadratic Trinomials
There are a few common methods for factorising quadratic trinomials. We'll focus on a popular and effective one that works well for expressions like 2x2 + 7x + 3.
Method: The "AC" Method (or Box Method)
This method is systematic and helps avoid common mistakes. Here's how it works:
- Identify the coefficients: In our expression 2x2 + 7x + 3, we have:
- a = 2 (the coefficient of x2)
- b = 7 (the coefficient of x)
- c = 3 (the constant term)
- Calculate 'a * c': Multiply the coefficient of the x2 term by the constant term.
a * c = 2 * 3 = 6
- Find two numbers that multiply to 'a * c' and add up to 'b': We need to find two numbers that, when multiplied together, give us 6, and when added together, give us 7. Let's list the factors of 6:
- 1 and 6 (1 * 6 = 6)
- 2 and 3 (2 * 3 = 6)
- 1 + 6 = 7
- 2 + 3 = 5
Success! The numbers are 1 and 6 because they multiply to 6 and add up to 7.
- Rewrite the middle term: Replace the middle term (7x) with the two numbers you found, each attached to an 'x'. So, 7x becomes 1x + 6x (or simply x + 6x).
Our expression now looks like: 2x2 + 1x + 6x + 3
- Group the terms: Group the first two terms together and the last two terms together.
(2x2 + 1x) + (6x + 3)
- Factor out the greatest common factor (GCF) from each group:
- From (2x2 + 1x): The GCF is 'x'. Factoring it out gives: x(2x + 1)
- From (6x + 3): The GCF is '3'. Factoring it out gives: 3(2x + 1)
Our expression is now: x(2x + 1) + 3(2x + 1)
- Factor out the common binomial: Notice that both parts have a common binomial factor: (2x + 1). Factor this out.
(2x + 1)(x + 3)
The Factorised Form
Therefore, the factorisation of 2x2 + 7x + 3 is (2x + 1)(x + 3).
You can always check your work by multiplying the factors back together using the FOIL method (First, Outer, Inner, Last):
- First: (2x) * (x) = 2x2
- Outer: (2x) * (3) = 6x
- Inner: (1) * (x) = 1x
- Last: (1) * (3) = 3
Adding these together: 2x2 + 6x + 1x + 3 = 2x2 + 7x + 3. It matches our original expression, so we know our factorisation is correct!
Frequently Asked Questions (FAQ)
How do I know which method to use for factorising?
For quadratic trinomials like 2x2 + 7x + 3, the "AC" method or the box method is very reliable. If the coefficient of x2 is 1 (like in x2 + 5x + 6), you can often find two numbers that multiply to the constant term and add to the middle term more directly. For more complex expressions, there are other techniques, but mastering these basic methods is a great start.
Why is factorising important in math?
Factorising is a fundamental skill in algebra. It allows you to simplify complex expressions, solve quadratic equations (by setting the factors equal to zero), graph quadratic functions, and work with rational expressions. It's a building block for many more advanced mathematical concepts.
What if I can't find two numbers that fit the criteria in step 3 of the AC method?
If you've followed the steps correctly and cannot find two numbers that multiply to 'a * c' and add to 'b', it means the quadratic expression cannot be factored using integers. Such expressions are called "prime" or "irreducible" over the integers.
Can I factorise 2x2 + 7x + 3 in a different order?
Yes, the order of the terms after rewriting the middle term (step 4) can vary. For instance, you could have written 2x2 + 6x + x + 3. If you group and factor these, you would get (2x2 + 6x) + (x + 3), which factors to 2x(x + 3) + 1(x + 3). This results in (2x + 1)(x + 3), the same answer. The final factored form will always be the same, regardless of the order you choose to rewrite the middle term.
