Which Function Is Equivalent To F(x) = X^2 - 7x + 10? A. F(x) = (x - 3)(x - 7) B. F(x) = (x + 7)(x + 3) C. F(x) = (x - 2)(x - 5) D. F(x) = (x - 2)(x + 5)
In the realm of mathematics, exploring the equivalence of functions is a fundamental concept. When we say two functions are equivalent, we mean they produce the same output for every input value. This equivalence can be expressed in different forms, such as the standard form (polynomial form) or the factored form. In this article, we will delve into the process of identifying the function that is equivalent to the quadratic function $f(x) = x^2 - 7x + 10$. We'll break down the steps involved, discuss the underlying mathematical principles, and ultimately reveal the correct equivalent function.
Understanding Quadratic Functions and Their Forms
Before we dive into the specific problem, let's establish a solid understanding of quadratic functions and their various forms. A quadratic function is a polynomial function of degree two, meaning the highest power of the variable (usually 'x') is 2. The standard form of a quadratic function is expressed as:
where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. In our given function, $f(x) = x^2 - 7x + 10$, we can identify a = 1, b = -7, and c = 10.
Another crucial form of a quadratic function is the factored form, which is expressed as:
where $r_1$ and $r_2$ are the roots or zeros of the quadratic function. The roots are the values of 'x' that make the function equal to zero (i.e., f(x) = 0). The factored form is particularly useful because it directly reveals the roots of the quadratic equation. To find the equivalent function in factored form, we need to determine the roots of the given quadratic function.
Factoring quadratic expressions is a core skill in algebra. The process involves reversing the distributive property (or the FOIL method) to express the quadratic as a product of two binomials. When we expand the factored form, we should arrive back at the standard form of the quadratic. Understanding the relationship between the coefficients in the standard form and the constants in the factored form is key to successful factoring. For instance, in the factored form $(x - r_1)(x - r_2)$, the constant term is the product of the roots ($r_1 * r_2$), and the coefficient of the 'x' term is the negative of the sum of the roots ($-(r_1 + r_2)$). This relationship provides a strategic way to find the correct factors.
Finding the Equivalent Function: A Step-by-Step Approach
Now, let's apply our understanding of quadratic functions and factoring to find the equivalent function to $f(x) = x^2 - 7x + 10$. Our goal is to rewrite this quadratic in its factored form. Here's the breakdown of the process:
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Identify the coefficients: In the given function, $f(x) = x^2 - 7x + 10$, we have a = 1, b = -7, and c = 10. These coefficients will guide our factoring process.
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Find two numbers that multiply to 'c' and add up to 'b': This is the core of factoring. We need to find two numbers that, when multiplied, give us 10 (the value of 'c') and, when added, give us -7 (the value of 'b'). Let's list the factor pairs of 10:
- 1 and 10
- -1 and -10
- 2 and 5
- -2 and -5
Out of these pairs, -2 and -5 satisfy both conditions: (-2) * (-5) = 10 and (-2) + (-5) = -7. These are the roots we are looking for.
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Write the factored form: Once we have the two numbers, we can directly write the factored form of the quadratic function. Since our numbers are -2 and -5, the factored form is:
This factored form represents the same quadratic function as $f(x) = x^2 - 7x + 10$ but in a different form.
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Verification: To ensure we have factored correctly, we can expand the factored form using the distributive property (or the FOIL method).
As we can see, expanding the factored form gives us back the original quadratic function, confirming that our factoring is correct.
Analyzing the Answer Choices
Now that we have determined the equivalent function in factored form, let's analyze the provided answer choices:
A. $f(x) = (x - 3)(x - 7)$ B. $f(x) = (x + 7)(x + 3)$ C. $f(x) = (x - 2)(x - 5)$ D. $f(x) = (x - 2)(x + 5)$
By comparing our factored form, $f(x) = (x - 2)(x - 5)$, with the answer choices, we can clearly see that option C is the correct equivalent function.
Let's briefly examine why the other options are incorrect:
- Option A: Expanding $(x - 3)(x - 7)$ would give us $x^2 - 10x + 21$, which is not the same as our original function.
- Option B: Expanding $(x + 7)(x + 3)$ would give us $x^2 + 10x + 21$, which is also incorrect.
- Option D: Expanding $(x - 2)(x + 5)$ would give us $x^2 + 3x - 10$, again not the same as the given function.
Therefore, the only correct answer is option C, $f(x) = (x - 2)(x - 5)$. This reinforces the importance of careful factoring and verification in solving mathematical problems.
Conclusion: The Power of Factoring
In this exploration, we successfully identified the function equivalent to $f(x) = x^2 - 7x + 10$ by leveraging the power of factoring. We demonstrated the step-by-step process of converting a quadratic function from its standard form to its factored form, highlighting the critical role of finding the roots of the equation.
Understanding factoring techniques is not just about manipulating algebraic expressions; it's about gaining a deeper insight into the structure and behavior of functions. The factored form provides valuable information about the roots of the function, which are crucial for solving equations, graphing functions, and understanding their real-world applications.
By mastering the concepts of quadratic functions and their various forms, you'll be well-equipped to tackle a wide range of mathematical challenges. This ability to manipulate and transform functions is a cornerstone of mathematical proficiency and opens doors to more advanced topics in algebra and beyond. Always remember to verify your results and embrace the beauty of mathematical equivalence in its various forms. The ability to connect different representations of the same function is a powerful tool in the mathematician's arsenal.