Simplify { (x-4)^2$}$.A. { X^2 - 8$}$B. { X^2 + 16$}$C. { X^2 - 16$}$D. { X^2 - 8x + 16$}$
Understanding the Concept of Squaring a Binomial
In mathematics, a binomial is an algebraic expression consisting of two terms. When we square a binomial, we multiply the expression by itself. This is a fundamental concept in algebra and is used extensively in various mathematical operations. In this article, we will simplify the expression {(x-4)^2$}$ and explore the different options available.
The Formula for Squaring a Binomial
The formula for squaring a binomial is:
{(a+b)^2 = a^2 + 2ab + b^2$}$
where {a$}$ and {b$}$ are the two terms in the binomial.
Applying the Formula to the Given Expression
In the given expression {(x-4)^2$}$, we can identify {a = x$}$ and {b = -4$}$. Applying the formula, we get:
{(x-4)^2 = x^2 + 2(x)(-4) + (-4)^2$}$
Simplifying the Expression
Now, let's simplify the expression by evaluating the terms:
{x^2 + 2(x)(-4) + (-4)^2 = x^2 - 8x + 16$}$
Comparing the Simplified Expression with the Options
Now that we have simplified the expression, let's compare it with the options available:
- Option A: {x^2 - 8$}$
- Option B: {x^2 + 16$}$
- Option C: {x^2 - 16$}$
- Option D: {x^2 - 8x + 16$}$
Conclusion
Based on our simplification, we can see that the correct answer is:
- Option D: {x^2 - 8x + 16$}$
This is because our simplified expression matches the option exactly.
Key Takeaways
- The formula for squaring a binomial is {(a+b)^2 = a^2 + 2ab + b^2$}$.
- When we square a binomial, we multiply the expression by itself.
- The simplified expression {(x-4)^2$}$ is {x^2 - 8x + 16$}$.
Practice Problems
To reinforce your understanding of the concept, try simplifying the following expressions:
- {(x+3)^2$}$
- {(x-2)^2$}$
- {(x+5)^2$}$
Real-World Applications
Squaring a binomial has numerous real-world applications in various fields, including:
- Physics: When calculating the distance traveled by an object under constant acceleration, we use the formula {s = ut + \frac{1}{2}at^2$}$, which involves squaring a binomial.
- Engineering: In designing electrical circuits, we use the formula {R = \frac{V^2}{P}$}$, which involves squaring a binomial.
- Computer Science: In programming, we use the formula {y = mx + b$}$, which involves squaring a binomial.
Conclusion
Frequently Asked Questions
In this article, we will address some of the most frequently asked questions related to simplifying the expression {(x-4)^2$}$.
Q: What is the formula for squaring a binomial?
A: The formula for squaring a binomial is {(a+b)^2 = a^2 + 2ab + b^2$}$.
Q: How do I apply the formula to the given expression?
A: To apply the formula, identify {a = x$}$ and {b = -4$}$ in the given expression {(x-4)^2$}$. Then, substitute these values into the formula and simplify.
Q: What is the simplified expression?
A: The simplified expression is {x^2 - 8x + 16$}$.
Q: Why is the correct answer {x^2 - 8x + 16$}$?
A: The correct answer is {x^2 - 8x + 16$}$ because it matches the simplified expression obtained by applying the formula.
Q: What are some real-world applications of squaring a binomial?
A: Squaring a binomial has numerous real-world applications in various fields, including physics, engineering, and computer science.
Q: How do I practice simplifying expressions?
A: To practice simplifying expressions, try simplifying the following expressions:
- {(x+3)^2$}$
- {(x-2)^2$}$
- {(x+5)^2$}$
Q: What are some common mistakes to avoid when simplifying expressions?
A: Some common mistakes to avoid when simplifying expressions include:
- Not applying the formula correctly
- Not evaluating the terms correctly
- Not simplifying the expression fully
Q: How do I check my work when simplifying expressions?
A: To check your work when simplifying expressions, compare your simplified expression with the correct answer. If they match, then your work is correct.
Q: What are some additional resources for learning about simplifying expressions?
A: Some additional resources for learning about simplifying expressions include:
- Online tutorials and videos
- Math textbooks and workbooks
- Online math communities and forums
Conclusion
In conclusion, simplifying the expression {(x-4)^2$}$ involves applying the formula for squaring a binomial and evaluating the terms. The correct answer is {x^2 - 8x + 16$}$. This concept has numerous real-world applications and is an essential tool in various mathematical operations. By practicing and reviewing the material, you can become proficient in simplifying expressions and applying the formula for squaring a binomial.
Practice Problems
To reinforce your understanding of the concept, try simplifying the following expressions:
- {(x+3)^2$}$
- {(x-2)^2$}$
- {(x+5)^2$}$
Real-World Applications
Squaring a binomial has numerous real-world applications in various fields, including:
- Physics: When calculating the distance traveled by an object under constant acceleration, we use the formula {s = ut + \frac{1}{2}at^2$}$, which involves squaring a binomial.
- Engineering: In designing electrical circuits, we use the formula {R = \frac{V^2}{P}$}$, which involves squaring a binomial.
- Computer Science: In programming, we use the formula {y = mx + b$}$, which involves squaring a binomial.
Conclusion
In conclusion, simplifying the expression {(x-4)^2$}$ involves applying the formula for squaring a binomial and evaluating the terms. The correct answer is {x^2 - 8x + 16$}$. This concept has numerous real-world applications and is an essential tool in various mathematical operations. By practicing and reviewing the material, you can become proficient in simplifying expressions and applying the formula for squaring a binomial.