Factor X 2 12x 35

Factoring the Quadratic Expression: x² + 12x + 35

This article delves into the complete process of factoring the quadratic expression x² + 12x + 35. We will explore various methods, from the straightforward trial-and-error approach to more systematic techniques, providing a comprehensive understanding for students of all levels. Understanding quadratic factoring is fundamental in algebra and serves as a building block for more advanced mathematical concepts. This guide will equip you with the knowledge and skills to tackle similar quadratic expressions confidently.

Introduction to Quadratic Expressions

A quadratic expression is a polynomial of degree two, meaning the highest power of the variable (usually x) is 2. It generally takes the form ax² + bx + c, where a, b, and c are constants. Factoring a quadratic expression means rewriting it as a product of two simpler expressions, usually two binomials. This process is crucial for solving quadratic equations and simplifying algebraic expressions. Our focus here is on factoring x² + 12x + 35.

Method 1: Trial and Error (Intuitive Approach)

This method relies on understanding how binomials multiply to produce a quadratic expression. We're looking for two binomials of the form (x + p)(x + q) that, when expanded using the FOIL method (First, Outer, Inner, Last), give us x² + 12x + 35.

Let's break down the FOIL method:

• First: x * x = x²
• Outer: x * q = qx
• Inner: p * x = px
• Last: p * q = pq

The sum of the Outer and Inner terms (qx + px) must equal 12x, and the Last term (pq) must equal 35.

We need to find two numbers (p and q) that add up to 12 and multiply to 35. Let's consider the factors of 35:

• 1 and 35
• 5 and 7

Since 5 + 7 = 12, we've found our numbers! Therefore, the factored form of x² + 12x + 35 is (x + 5)(x + 7).

To verify, let's expand (x + 5)(x + 7) using FOIL:

• First: x * x = x²
• Outer: x * 7 = 7x
• Inner: 5 * x = 5x
• Last: 5 * 7 = 35

Combining like terms, we get x² + 7x + 5x + 35 = x² + 12x + 35. Our factoring is correct!

Method 2: The AC Method (Systematic Approach)

The AC method provides a more structured approach, especially helpful when dealing with more complex quadratic expressions. This method is particularly useful when the coefficient of x² (a) is not equal to 1. Even though our coefficient a is 1 here, understanding the AC method is valuable for future problems.

1. Identify a, b, and c: In our expression x² + 12x + 35, a = 1, b = 12, and c = 35.

2. Find the product ac: ac = 1 * 35 = 35

3. Find two numbers that add up to b and multiply to ac: We need two numbers that add up to 12 and multiply to 35. As we found in the trial-and-error method, these numbers are 5 and 7.

4. Rewrite the expression: Rewrite the middle term (12x) as the sum of these two numbers multiplied by x: x² + 5x + 7x + 35

5. Factor by grouping: Group the terms in pairs and factor out the greatest common factor (GCF) from each pair:

   x(x + 5) + 7(x + 5)

6. Factor out the common binomial: Notice that (x + 5) is common to both terms. Factor it out:

   (x + 5)(x + 7)

Again, we arrive at the factored form (x + 5)(x + 7).

Method 3: Completing the Square (Advanced Technique)

Completing the square is a more advanced technique used primarily in solving quadratic equations but can also be applied to factoring. While less efficient for this specific problem, it's a valuable tool to understand.

1. Move the constant term to the right side: x² + 12x = -35

2. Take half of the coefficient of x, square it, and add it to both sides: Half of 12 is 6, and 6² = 36. So we add 36 to both sides:

   x² + 12x + 36 = -35 + 36

3. Factor the left side as a perfect square trinomial: The left side is now a perfect square trinomial: (x + 6)².

   (x + 6)² = 1

4. Take the square root of both sides: x + 6 = ±√1 (remember the ± because both 1 and -1 squared equal 1)

5. Solve for x: x = -6 ± 1, so x = -5 or x = -7.

While this method gives us the roots (-5 and -7), to get the factored form, we need to rewrite the equation as (x + 5)(x + 7) = 0. Notice how the roots are the opposite of the constants in the factored form.

Understanding the Relationship Between Roots and Factors

The roots of a quadratic equation (the values of x that make the equation equal to zero) are directly related to its factors. In our case, the roots are -5 and -7. The factors are (x + 5) and (x + 7). The relationship is that if r is a root, then (x - r) is a factor.

Solving Quadratic Equations Using Factoring

Once we've factored the quadratic expression, we can use it to solve the corresponding quadratic equation. For example, to solve x² + 12x + 35 = 0, we use the factored form:

(x + 5)(x + 7) = 0

This equation is true if either (x + 5) = 0 or (x + 7) = 0. Therefore, the solutions (roots) are x = -5 and x = -7.

Expanding to More Complex Quadratics

The techniques described above, particularly the AC method, are applicable to more complex quadratic expressions where the coefficient of x² is not 1. For example, consider the expression 2x² + 11x + 12. The AC method will guide you efficiently through the factoring process.

Frequently Asked Questions (FAQ)

Q: What if the quadratic expression cannot be factored?

A: Not all quadratic expressions can be factored using integers. In such cases, you might need to use the quadratic formula to find the roots, or you can leave the expression in its unfactored form.

Q: Is there a single "best" method for factoring quadratics?

A: The best method depends on your comfort level and the specific quadratic expression. Trial and error is quick for simpler expressions, while the AC method is more systematic for complex ones.

Q: What are some common mistakes to avoid when factoring?

A: Common mistakes include incorrect signs, forgetting to check your work by expanding the factored form, and not considering all possible factor pairs. Always verify your answer by expanding!

Q: How can I improve my factoring skills?

A: Practice is key! Work through numerous examples, starting with simpler expressions and gradually increasing the difficulty. Understanding the underlying principles of the different methods is crucial for long-term success.

Conclusion

Factoring quadratic expressions like x² + 12x + 35 is a fundamental skill in algebra. This article has explored various methods—trial and error, the AC method, and completing the square—providing a comprehensive understanding of the process. Mastering these techniques is essential for solving quadratic equations, simplifying algebraic expressions, and progressing to more advanced mathematical concepts. Remember that practice is crucial, and by understanding the underlying principles, you'll build confidence and proficiency in factoring quadratic expressions. Don't hesitate to revisit these methods and practice regularly to reinforce your learning.