To factorise a quadratic equation by splitting the middle term, you follow a series of steps focused on breaking down the linear term to reveal common factors that allow you to represent the quadratic in factored form. This method is particularly useful when the quadratic equation is in the form ax² + bx + c = 0.
Understanding the Process
The goal is to rewrite the quadratic equation in a way that it can be easily factored into two binomials. Here's the systematic approach:
Steps for Splitting the Middle Term
Here's how to factor a quadratic equation by splitting the middle term, based on the information in the reference:
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Identify the coefficients: Consider the quadratic equation in the standard form: ax² + bx + c = 0. Identify the values of a, b, and c.
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Find two numbers: This is the key step. You need to find two numbers whose product equals ac (the product of the coefficient of x² and the constant term), and whose sum equals b (the coefficient of x).
- Product = ac
- Sum = b
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Split the middle term: Replace the middle term (bx) with the sum of two terms, using the two numbers you just found. So, if the numbers are m and n, you would replace bx with mx + nx. Now you have an equation with four terms.
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Factor by grouping: Group the terms in pairs and factor out the common factor from each pair.
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Simplify: If done correctly, you'll have a common binomial factor. Factor out this common binomial, and you'll have factored the quadratic equation.
Detailed Breakdown with Examples
Let's illustrate with examples:
Example 1: Factor x² + 5x + 6
- Identify: a = 1, b = 5, c = 6
- Find two numbers: We need two numbers that multiply to (1)(6) = 6 and add to 5. The numbers are 2 and 3.
- Split the middle term: x² + 2x + 3x + 6
- Factor by grouping: x(x + 2) + 3(x + 2)
- Simplify: (x + 2)(x + 3)
Therefore, x² + 5x + 6 factors to (x + 2)(x + 3).
Example 2: Factor 2x² + 7x + 3
- Identify: a = 2, b = 7, c = 3
- Find two numbers: We need two numbers that multiply to (2)(3) = 6 and add to 7. The numbers are 6 and 1.
- Split the middle term: 2x² + 6x + 1x + 3
- Factor by grouping: 2x(x + 3) + 1(x + 3)
- Simplify: (x + 3)(2x + 1)
Thus, 2x² + 7x + 3 factors to (x + 3)(2x + 1).
When the Method Works Best
This method is very useful for quadratic equations where the roots are rational numbers. However, it may not be the most efficient for all quadratic equations (especially those with irrational roots). In such cases, other methods like completing the square or using the quadratic formula may be more appropriate.
In summary, splitting the middle term involves systematically finding two numbers to rewrite the linear term, facilitating factoring by grouping to reveal the factored form of the quadratic equation.
