Factorizing Quadratic Expressions: Techniques and Applications
When dealing with quadratic expressions, it is crucial to understand the various techniques available to correctly factorize them. One such expression is c2 - 13c - 42. In this article, we will explore the most appropriate methods to factorize this expression and others, with a focus on practical applications and common pitfalls.
Understanding Quadratic Equations and Factorization
A quadratic equation in the form ax2 bx c 0 can often be factored into the form (px q)(rx s) 0. The key is to find the correct values of p, q, r, and s that satisfy the conditions of the equation.
Using Vieta's Theorem
Vieta's theorem helps us find the roots of the quadratic equation without directly factoring the expression. For the equation c2 - 13c - 42, the roots can be found by solving the quadratic formula:
c (13 ± √(132 - 4(1)(-42))) / 2(1)
Simplifying, we get:
c (13 ± √(169 168)) / 2
c (13 ± √337) / 2
The roots are approximately:
c ≈ 16.195, c ≈ -2.195
Factorization Techniques
If we want to factor the expression without using the quadratic formula, we can look for two numbers whose product is ac -42 and whose sum is b -13. These numbers can be found by listing the factors of -42:
1 and -42 -1 and 42 2 and -21 -2 and 21 3 and -14 -3 and 14 6 and -7 -6 and 7The pairs that add up to -13 are -6 and -7. Therefore, the factorization of the expression is:
c2 - 13c - 42 (c - 6)(c - 7)
Mid-Term Breaking
Another method to factorize c2 - 13c - 42 is to break the middle term. We can rewrite the expression as:
c2 - 6c - 7c - 42
Grouping the terms, we get:
(c2 - 6c) - (7c 42)
Factoring out common terms:
c(c - 6) - 7(c - 6)
Factoring out the common factor (c - 6):
(c - 6)(c - 7)
Application of Factorization
Factoring quadratic expressions is crucial in many applications, such as solving polynomial equations, simplifying functions, and understanding the behavior of parabolas. For example, if we want to find the roots of the equation:
c2 - 13c - 42 0
We can use the factorization:
(c - 6)(c - 7) 0
Setting each factor equal to zero:
c - 6 0
c - 7 0
The roots are:
c 6 and c 7
Conclusion
Understanding and applying the appropriate methods of factorization is essential for solving quadratic equations. The process of factorizing expressions like c2 - 13c - 42 can be approached through various techniques, including Vieta's theorem, mid-term breaking, and seeking the correct factors. This knowledge is crucial for a wide range of mathematical and real-world applications.