2x2+21x+49

2025

2 min read 30-03-2025

This article explores how to factor the quadratic expression 2x² + 21x + 49. Factoring quadratics is a fundamental skill in algebra, crucial for solving equations and simplifying expressions. We'll walk through the process step-by-step, demonstrating different methods and explaining the underlying concepts. Understanding how to factor this specific expression will help you approach similar problems with confidence.

Understanding Quadratic Expressions

Before we delve into factoring 2x² + 21x + 49, let's briefly review quadratic expressions. A quadratic expression is a polynomial of degree two, meaning the highest power of the variable (x in this case) is 2. It generally takes the form ax² + bx + c, where a, b, and c are constants. Our expression, 2x² + 21x + 49, fits this form perfectly, with a = 2, b = 21, and c = 49.

Method 1: Factoring by Grouping

One common method for factoring quadratics is factoring by grouping. This method is particularly useful when the quadratic doesn't readily factor using simpler techniques. Unfortunately, 2x² + 21x + 49 doesn't easily factor using this method. Let's explore why. To use this method we would need to find two numbers that add up to 'b' (21 in our case) and multiply to 'ac' (249 = 98). Finding those two numbers proves difficult.

Method 2: The AC Method (for Trinomials)

Another powerful method is the AC method. This is a variation on factoring by grouping that is often more efficient. Let's apply it to 2x² + 21x + 49.

Multiply a and c: 2 * 49 = 98
Find two numbers: We need two numbers that add up to 21 (our 'b' value) and multiply to 98. These numbers are 7 and 14 (7 + 14 = 21 and 7 * 14 = 98).
Rewrite the middle term: We rewrite the middle term (21x) as the sum of these two numbers: 2x² + 7x + 14x + 49
Factor by grouping: Now, we group the terms and factor: (2x² + 7x) + (14x + 49) = x(2x + 7) + 7(2x + 7)
Factor out the common binomial: Notice that (2x + 7) is common to both terms. We factor it out: (2x + 7)(x + 7)

Therefore, the factored form of 2x² + 21x + 49 is (2x + 7)(x + 7).

Verification

To verify our factoring is correct, we can expand the factored form using the FOIL method (First, Outer, Inner, Last):

(2x + 7)(x + 7) = 2x² + 14x + 7x + 49 = 2x² + 21x + 49

This matches our original expression, confirming that our factoring is accurate.

Solving Quadratic Equations

Knowing how to factor a quadratic expression like 2x² + 21x + 49 is crucial for solving quadratic equations. If you were given the equation 2x² + 21x + 49 = 0, you could use the factored form (2x + 7)(x + 7) = 0 to find the solutions: x = -7/2 and x = -7.

Conclusion

Factoring quadratic expressions is a fundamental algebraic skill. While the expression 2x² + 21x + 49 might seem challenging at first, applying the AC method allows us to systematically factor it into (2x + 7)(x + 7). Understanding these methods empowers you to tackle similar problems efficiently and accurately. Remember to always verify your factoring by expanding the factored expression to ensure it matches the original.