Factoring Quadratic Expressions A Step-by-Step Guide To Factoring X² + 12x + 36
Factoring quadratic expressions is a fundamental skill in algebra, enabling us to simplify expressions, solve equations, and analyze mathematical relationships. The given expression, x² + 12x + 36, is a quadratic trinomial, which means it's a polynomial with three terms and the highest power of the variable being 2. Our goal is to rewrite this expression as a product of two binomials. This process, known as factoring, is essentially the reverse of expanding binomials using the distributive property (also known as the FOIL method).
Understanding Quadratic Trinomials: A quadratic trinomial generally takes the form ax² + bx + c, where a, b, and c are constants. In our case, a = 1, b = 12, and c = 36. Factoring such expressions involves finding two numbers that satisfy specific conditions related to the coefficients b and c. Specifically, we need to find two numbers that multiply to c (36 in our case) and add up to b (12 in our case).
Methods of Factoring: There are several methods for factoring quadratic trinomials, but the most common approach involves identifying the factors of the constant term (c) and then checking if any pair of factors adds up to the coefficient of the linear term (b). Another method is recognizing perfect square trinomials, which can be factored directly into the square of a binomial. We will explore both approaches in the context of our expression.
Identifying the Pattern
Before diving into specific factoring techniques, it's crucial to identify patterns and characteristics of the given expression. This initial observation can significantly streamline the factoring process. In the case of x² + 12x + 36, notice that the first term (x²) is a perfect square, and the last term (36) is also a perfect square (6²). This suggests that the expression might be a perfect square trinomial, a special type of quadratic trinomial that factors into the square of a binomial. Recognizing such patterns can save time and effort in the factoring process.
A perfect square trinomial is a trinomial that can be expressed in the form (ax + b)² or (ax - b)². When expanded, these expressions result in a²x² + 2abx + b² and a²x² - 2abx + b², respectively. Our expression, x² + 12x + 36, closely resembles the form a²x² + 2abx + b². If we can find values for a and b that fit this pattern, we can directly factor the expression without resorting to trial and error methods.
Recognizing a Perfect Square Trinomial
To confirm whether x² + 12x + 36 is a perfect square trinomial, let's compare it to the general form a²x² + 2abx + b². Here, a²x² corresponds to x², which means a = 1. Similarly, b² corresponds to 36, implying that b could be either 6 or -6. We need to check if the middle term, 12x, matches the form 2abx. If we take b = 6, then 2abx becomes 2 * 1 * 6 * x = 12x, which matches the middle term of our expression.
Since a = 1 and b = 6 satisfy the conditions for a perfect square trinomial, we can confidently conclude that x² + 12x + 36 is indeed a perfect square trinomial. This realization simplifies the factoring process significantly. Instead of searching for two numbers that multiply to 36 and add up to 12, we can directly apply the perfect square trinomial pattern.
The general form of a factored perfect square trinomial is (ax + b)² or (ax - b)². In our case, since the middle term of the trinomial is positive, we will use the form (ax + b)². Substituting a = 1 and b = 6, we get (x + 6)². This means that the factored form of x² + 12x + 36 is (x + 6)(x + 6).
Factoring the Expression
Now that we've recognized the pattern of a perfect square trinomial, the factoring process becomes straightforward. We know that x² + 12x + 36 can be written in the form (x + 6)², which is equivalent to (x + 6)(x + 6). This means that the expression factors into two identical binomials, both being (x + 6).
To further illustrate this, let's consider the steps involved in expanding (x + 6)(x + 6) using the distributive property (or the FOIL method): (x + 6)(x + 6) = x(x + 6) + 6(x + 6) = x² + 6x + 6x + 36 = x² + 12x + 36. This confirms that our factored form, (x + 6)(x + 6), is indeed equivalent to the original expression, x² + 12x + 36.
Therefore, the factored form of the expression x² + 12x + 36 is (x + 6)(x + 6). This result aligns with option A in the given choices.
Verification and Alternative Approaches
While we've successfully factored the expression by recognizing it as a perfect square trinomial, it's always beneficial to verify our result and explore alternative approaches. This not only reinforces our understanding but also equips us with different problem-solving strategies.
Verification: To verify our factored form, (x + 6)(x + 6), we can expand it again using the distributive property. As we demonstrated earlier, expanding (x + 6)(x + 6) results in x² + 12x + 36, which is the original expression. This confirms that our factoring is correct.
Alternative Approach (Trial and Error): If we hadn't recognized the perfect square trinomial pattern, we could have used a trial-and-error approach. This involves finding two numbers that multiply to 36 and add up to 12. The factors of 36 are: 1 and 36, 2 and 18, 3 and 12, 4 and 9, and 6 and 6. Among these pairs, only 6 and 6 add up to 12. Therefore, the expression can be factored as (x + 6)(x + 6). This approach, while sometimes more time-consuming, is a valuable alternative when pattern recognition is not immediately apparent.
Conclusion
In conclusion, the expression x² + 12x + 36 can be factored as (x + 6)(x + 6). This was achieved by recognizing the expression as a perfect square trinomial and applying the corresponding factoring pattern. We also verified our result by expanding the factored form and demonstrated an alternative trial-and-error approach. Mastering factoring techniques is crucial for success in algebra and higher-level mathematics.
The correct answer is A. (x+6)(x+6). Understanding factoring techniques, especially recognizing patterns like perfect square trinomials, is fundamental for algebraic manipulation and problem-solving.
