Factoring The Expression 7c² - 4c - 20 A Step-by-Step Guide

Factoring quadratic expressions is a fundamental skill in algebra. This article will guide you through the process of factoring the expression 7c² - 4c - 20, a quadratic trinomial. We'll explore the steps involved, explain the underlying principles, and provide a clear, step-by-step solution to help you understand the factorization process.

Understanding Quadratic Expressions

Before diving into the factorization, let's understand what a quadratic expression is. A quadratic expression is a polynomial of degree two, generally written in the form ax² + bx + c, where a, b, and c are constants and a is not equal to zero. In our case, the expression 7c² - 4c - 20 fits this form, with a = 7, b = -4, and c = -20. Factoring a quadratic expression involves breaking it down into a product of two binomials. This is the reverse process of expanding two binomials using the distributive property (also known as FOIL - First, Outer, Inner, Last).

Why is factoring important? Factoring quadratic expressions is crucial for solving quadratic equations, simplifying algebraic fractions, and understanding the behavior of quadratic functions. It's a cornerstone of algebra and has applications in various fields like physics, engineering, and computer science.

When you encounter a quadratic expression, your goal is to find two binomials that, when multiplied together, give you the original expression. There are several techniques for factoring, including the trial-and-error method, the grouping method, and using the quadratic formula. We'll focus on the trial-and-error method and a modified grouping method suitable for this expression.

Identifying the Structure of the Factored Form

We are given a partially factored form: (7c + [?])(c - []). This structure provides us with valuable information. The first term in the first binomial is 7c, and the first term in the second binomial is c. This aligns with the leading term of the quadratic expression, 7c², as 7c multiplied by c equals 7c². Our task is to find the missing constants that, when inserted into the blanks, will complete the factorization. These constants must satisfy specific conditions related to the coefficients of the original quadratic expression.

The factored form implies that the original quadratic expression can be written as a product of two binomials. These binomials are linear expressions in terms of the variable c. The constants we are looking for represent the y-intercepts of these linear expressions when graphed. Furthermore, the product of these constants must contribute to the constant term of the quadratic expression, which is -20 in our case. Additionally, the sum of the cross-products of these constants with the coefficients of c in the binomials must equal the coefficient of the linear term in the quadratic expression, which is -4.

This partially factored form simplifies our task. Instead of trying all possible combinations, we can focus on finding two numbers that, when placed in the blanks, satisfy the conditions imposed by the coefficients of the quadratic expression. The coefficient 7 in front of the c term in the first binomial means that one of the factors of -20 will be multiplied by 7 when we expand the product. This will significantly impact our selection of factors.

The Factoring Process: A Step-by-Step Guide

Now, let's delve into the process of factoring 7c² - 4c - 20 using the given structure (7c + [?])(c - []). We will employ a trial-and-error approach, guided by the relationships between the coefficients of the quadratic expression and the constants in the factored form.

Step 1: Identify the factors of the leading coefficient and the constant term.

• The leading coefficient is 7, which has factors 1 and 7.
• The constant term is -20, which has factor pairs: (1, -20), (-1, 20), (2, -10), (-2, 10), (4, -5), and (-4, 5).

Step 2: Consider the given binomial structure (7c + [?])(c - []).

Since the first term of the first binomial is 7c and the first term of the second binomial is c, we know that when the binomials are multiplied, the 7c² term will be correctly produced. Now we need to find the two constants that fit into the blanks. Let's call these constants m and n. Our factored form will look like (7c + m)(c + n), but remember one of m or n will be negative.

Step 3: Determine the conditions the constants must satisfy.

When we expand (7c + m)(c + n), we get:

7c² + 7nc + mc + mn

7c² + (7n + m)c + mn

Comparing this to our original expression, 7c² - 4c - 20, we can establish the following conditions:

• mn = -20 (The product of the constants must equal the constant term)
• 7n + m = -4 (The sum of the cross-products must equal the coefficient of the linear term)

Step 4: Trial and Error with Factor Pairs

Now we will test the factor pairs of -20, considering the constraint 7n + m = -4.

• Try (4, -5): If m = 4 and n = -5, then 7n + m = 7(-5) + 4 = -35 + 4 = -31. This does not equal -4.
• Try (-4, 5): If m = -4 and n = 5, then 7n + m = 7(5) + (-4) = 35 - 4 = 31. This does not equal -4.
• Try (5, -4): If m = 5 and n = -4, then 7n + m = 7(-4) + 5 = -28 + 5 = -23. This does not equal -4.
• Try (-5, 4): If m = -5 and n = 4, then 7n + m = 7(4) + (-5) = 28 - 5 = 23. This does not equal -4.
• Try (10, -2): If m = 10 and n = -2, then 7n + m = 7(-2) + 10 = -14 + 10 = -4. This works!

Step 5: Write the factored expression.

Since m = 10 and n = -2 satisfy the conditions, we can write the factored expression as:

(7c + 10)(c - 2)

Step 6: Verify the factorization.

To verify our factorization, we can expand the binomials:

(7c + 10)(c - 2) = 7c² - 14c + 10c - 20 = 7c² - 4c - 20

This matches our original expression, so our factorization is correct.

Solution

The factored form of 7c² - 4c - 20 is (7c + 10)(c - 2).

Therefore, the missing values are 10 and 2.

Tips for Factoring Quadratic Expressions

Factoring quadratic expressions can sometimes be challenging, but here are some helpful tips:

• Always look for a common factor first: Before attempting any other factoring method, check if there's a greatest common factor (GCF) that can be factored out from all terms.
• Understand the relationships between coefficients and constants: The constant term in the quadratic expression is the product of the constants in the factored binomials, and the coefficient of the linear term is related to the sum of the cross-products.
• Practice makes perfect: The more you practice factoring, the better you'll become at recognizing patterns and applying the appropriate techniques.
• Use the trial-and-error method systematically: When using the trial-and-error method, try different factor pairs in an organized manner to avoid missing a correct combination.
• Verify your factorization: Always verify your factored expression by expanding it to ensure it matches the original quadratic expression.
• Consider alternative methods: If trial and error proves difficult, explore other methods like the grouping method or the quadratic formula.

Conclusion

Factoring the quadratic expression 7c² - 4c - 20 involves finding two binomials whose product equals the original expression. By understanding the relationships between the coefficients and constants, we can systematically find the correct factors. In this case, the factored form is (7c + 10)(c - 2). Factoring is a vital skill in algebra, with applications across various mathematical and scientific disciplines. Mastering this skill will significantly enhance your problem-solving abilities in algebra and beyond.

Through this comprehensive guide, we've demonstrated how to factor a specific quadratic expression, highlighted the underlying principles, and provided practical tips for tackling similar problems. Remember that consistent practice and a solid understanding of the concepts are key to mastering factoring quadratic expressions. Keep practicing, and you'll become more confident and efficient in your factoring skills. The ability to factor quadratic expressions opens doors to more advanced mathematical concepts and applications, making it a worthwhile skill to cultivate.