Which Expression Is Equivalent To \[$(5-t)^2\$\], If \[$t^2=2\$\]?A. \[$25 - 10t\$\] B. \[$25 + 10t\$\] C. \[$27 - 10t\$\] D. \[$27\$\]

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore the process of simplifying algebraic expressions, with a focus on the given problem: finding an equivalent expression to {(5-t)^2$}$, given that {t^2=2$}$. We will break down the solution into manageable steps, making it easy to follow and understand.

Understanding the Problem

The problem asks us to find an equivalent expression to {(5-t)^2$}$, given that {t^2=2$}$. To solve this problem, we need to use the concept of algebraic expansion and simplification.

Step 1: Expand the Expression

The first step is to expand the given expression {(5-t)^2$}$ using the formula (a−b)2=a2−2ab+b2{(a-b)^2 = a^2 - 2ab + b^2}. In this case, {a=5$}$ and {b=t$}$.

{(5-t)^2 = 5^2 - 2(5)(t) + t^2$}$

Step 2: Simplify the Expression

Now that we have expanded the expression, we can simplify it by substituting the value of {t^2=2$}$.

{(5-t)^2 = 25 - 10t + 2$}$

Step 3: Combine Like Terms

The next step is to combine like terms in the expression.

{(5-t)^2 = 27 - 10t$}$

Conclusion

In conclusion, the equivalent expression to {(5-t)^2$}$, given that {t^2=2$}$, is ${27 - 10t\$}. This solution demonstrates the importance of algebraic expansion and simplification in solving mathematical problems.

Answer

The correct answer is:

  • A. ${25 - 10t\$} is incorrect, as it does not take into account the value of {t^2=2$}$.
  • B. ${25 + 10t\$} is incorrect, as it does not take into account the value of {t^2=2$}$.
  • C. ${27 - 10t\$} is the correct answer, as it correctly simplifies the expression using the value of {t^2=2$}$.
  • D. ${27\$} is incorrect, as it does not take into account the value of {t^2=2$}$.

Final Thoughts

Simplifying algebraic expressions is an essential skill for any math enthusiast. By following the steps outlined in this article, you can simplify complex expressions and arrive at the correct solution. Remember to always expand and simplify expressions using the correct formulas and values.

Additional Resources

For more information on algebraic expressions and simplification, check out the following resources:

  • Khan Academy: Algebraic Expressions
  • Mathway: Algebraic Expression Simplifier
  • Wolfram Alpha: Algebraic Expression Solver

Frequently Asked Questions

Q: What is the equivalent expression to {(5-t)^2$}$, given that {t^2=2$}$?

A: The equivalent expression is ${27 - 10t\$}.

Q: How do I simplify algebraic expressions?

A: To simplify algebraic expressions, follow these steps:

  1. Expand the expression using the correct formulas.
  2. Simplify the expression by combining like terms.
  3. Use the correct values and formulas to arrive at the correct solution.

Q: What are some common algebraic expressions?

A: Some common algebraic expressions include:

  • {(a-b)^2 = a^2 - 2ab + b^2}$
  • {(a+b)^2 = a^2 + 2ab + b^2}$
  • {(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3}$

Q: How do I use algebraic expressions in real-life situations?

A: Algebraic expressions are used in a variety of real-life situations, including:

  • Science: Algebraic expressions are used to model and solve scientific problems.
  • Engineering: Algebraic expressions are used to design and optimize systems.
  • Finance: Algebraic expressions are used to calculate interest rates and investment returns.

Conclusion

Q: What is an algebraic expression?

A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations. It is a way to represent a mathematical relationship between variables and constants.

Q: What are some common types of algebraic expressions?

A: Some common types of algebraic expressions include:

  • Polynomial expressions: These are expressions that consist of variables and constants, and are typically written in the form of a sum of terms, where each term is a product of a variable and a constant.
  • Rational expressions: These are expressions that consist of a fraction of two polynomials.
  • Exponential expressions: These are expressions that consist of a base raised to a power.

Q: How do I simplify algebraic expressions?

A: To simplify algebraic expressions, follow these steps:

  1. Expand the expression: Use the distributive property to expand the expression.
  2. Combine like terms: Combine terms that have the same variable and exponent.
  3. Simplify the expression: Simplify the expression by canceling out any common factors.

Q: What is the order of operations?

A: The order of operations is a set of rules that dictate the order in which mathematical operations should be performed. The order of operations is:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I evaluate algebraic expressions?

A: To evaluate algebraic expressions, follow these steps:

  1. Substitute the values: Substitute the given values into the expression.
  2. Simplify the expression: Simplify the expression by combining like terms and canceling out any common factors.
  3. Evaluate the expression: Evaluate the expression by performing any remaining operations.

Q: What are some common algebraic expressions?

A: Some common algebraic expressions include:

  • Linear expressions: These are expressions that consist of a variable and a constant, and are typically written in the form of ax + b.
  • Quadratic expressions: These are expressions that consist of a variable squared and a constant, and are typically written in the form of ax^2 + bx + c.
  • Polynomial expressions: These are expressions that consist of a sum of terms, where each term is a product of a variable and a constant.

Q: How do I use algebraic expressions in real-life situations?

A: Algebraic expressions are used in a variety of real-life situations, including:

  • Science: Algebraic expressions are used to model and solve scientific problems.
  • Engineering: Algebraic expressions are used to design and optimize systems.
  • Finance: Algebraic expressions are used to calculate interest rates and investment returns.

Q: What are some common applications of algebraic expressions?

A: Some common applications of algebraic expressions include:

  • Graphing: Algebraic expressions are used to graph functions and relationships.
  • Optimization: Algebraic expressions are used to optimize systems and processes.
  • Modeling: Algebraic expressions are used to model real-world phenomena.

Q: How do I choose the correct algebraic expression?

A: To choose the correct algebraic expression, follow these steps:

  1. Identify the problem: Identify the problem you are trying to solve.
  2. Choose the correct expression: Choose the algebraic expression that best models the problem.
  3. Simplify the expression: Simplify the expression by combining like terms and canceling out any common factors.

Conclusion

In conclusion, algebraic expressions are a fundamental concept in mathematics, and are used to model and solve a wide range of problems. By understanding how to simplify and evaluate algebraic expressions, you can apply them to real-life situations and make informed decisions.