How to Factor 2x² + 7x + 3
Factoring quadratic expressions is a fundamental skill in algebra. This guide will walk you through factoring the expression 2x² + 7x + 3 step-by-step. We'll explore different methods and highlight key considerations to help you master this technique.
Understanding Quadratic Expressions
Before diving into the factoring process, let's refresh our understanding. A quadratic expression is an expression of the form ax² + bx + c, where a, b, and c are constants, and a ≠ 0. In our case, a = 2, b = 7, and c = 3.
Method 1: AC Method
The AC method is a systematic approach to factoring quadratic expressions. Here's how it works for 2x² + 7x + 3:
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Find the product 'ac': Multiply the coefficient of the x² term (a) by the constant term (c). In this case, ac = 2 * 3 = 6.
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Find two numbers that add up to 'b' and multiply to 'ac': We need two numbers that add up to 7 (the coefficient of the x term) and multiply to 6. These numbers are 6 and 1 (6 + 1 = 7 and 6 * 1 = 6).
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Rewrite the middle term: Rewrite the middle term (7x) using the two numbers we found: 2x² + 6x + 1x + 3
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Factor by grouping: Group the terms in pairs and factor out the greatest common factor (GCF) from each pair:
2x(x + 3) + 1(x + 3)
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Factor out the common binomial: Notice that (x + 3) is a common factor in both terms. Factor it out:
(x + 3)(2x + 1)
Therefore, the factored form of 2x² + 7x + 3 is (x + 3)(2x + 1).
Method 2: Trial and Error
This method involves trying different combinations of factors until you find the correct one. It can be quicker for simpler quadratics but may take more time for complex expressions.
For 2x² + 7x + 3, we look for two binomials that multiply to give the original quadratic. Since the first term is 2x², we know the first terms of the binomials must be 2x and x (or x and 2x). The last term is 3, so the last terms of the binomials could be 1 and 3 or 3 and 1. By trying different combinations, we find that (x + 3)(2x + 1) works. This is because when we expand this using the FOIL method (First, Outer, Inner, Last), we get back to the original expression 2x² + 7x + 3.
Checking Your Answer
Always check your answer by expanding the factored form using the FOIL method (First, Outer, Inner, Last) or distribution. If you get back to the original quadratic expression, you know you have factored correctly.
Practice Makes Perfect
Factoring quadratic expressions takes practice. The more you practice, the faster and more proficient you'll become. Try factoring other quadratic expressions using the methods described above. Remember to always check your work!
This comprehensive guide should help you understand and factor 2x² + 7x + 3 with confidence. Remember to choose the method you find most comfortable and efficient.
