Find The Value Of The Expression $2a^2 - 2a$ When $a = \frac{1}{4}$.

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will explore how to find the value of a given algebraic expression when a variable is assigned a specific value. We will use a step-by-step approach to simplify the expression and arrive at the final answer.

Understanding the Problem

The problem requires us to find the value of the expression $2a^2 - 2a$ when $a = \frac{1}{4}$. To solve this problem, we need to substitute the value of $a$ into the expression and simplify it.

Step 1: Substitute the Value of a

The first step is to substitute the value of $a$ into the expression. We are given that $a = \frac{1}{4}$. Substituting this value into the expression, we get:

2(14)2−2(14)2\left(\frac{1}{4}\right)^2 - 2\left(\frac{1}{4}\right)

Step 2: Simplify the Expression

Now that we have substituted the value of $a$, we need to simplify the expression. To do this, we need to follow the order of operations (PEMDAS):

  1. Evaluate the expressions inside the parentheses.
  2. Exponentiate the results.
  3. Multiply and divide from left to right.
  4. Add and subtract from left to right.

Let's simplify the expression step by step:

  1. Evaluate the expressions inside the parentheses:

(14)2=116\left(\frac{1}{4}\right)^2 = \frac{1}{16}

2(14)=122\left(\frac{1}{4}\right) = \frac{1}{2}

  1. Exponentiate the results:

2(116)=182\left(\frac{1}{16}\right) = \frac{1}{8}

  1. Multiply and divide from left to right:

18−12\frac{1}{8} - \frac{1}{2}

  1. Add and subtract from left to right:

18−12=−38\frac{1}{8} - \frac{1}{2} = -\frac{3}{8}

Conclusion

In this article, we have shown how to find the value of the expression $2a^2 - 2a$ when $a = \frac{1}{4}$. We followed a step-by-step approach to simplify the expression and arrive at the final answer. By substituting the value of $a$ and simplifying the expression, we arrived at the final answer of $-\frac{3}{8}$.

Tips and Tricks

  • When substituting the value of a variable into an expression, make sure to follow the order of operations (PEMDAS).
  • Simplify the expression step by step, following the order of operations.
  • Use a calculator or a computer algebra system to check your answer.

Real-World Applications

Solving algebraic expressions is a crucial skill in many real-world applications, such as:

  • Science: Algebraic expressions are used to model real-world phenomena, such as the motion of objects and the behavior of populations.
  • Engineering: Algebraic expressions are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Algebraic expressions are used to model economic systems and make predictions about future trends.

Common Mistakes

  • Failing to follow the order of operations (PEMDAS).
  • Not simplifying the expression step by step.
  • Not checking the answer using a calculator or a computer algebra system.

Conclusion

Q: What is an algebraic expression?

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

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when simplifying an algebraic expression. The acronym PEMDAS stands for:

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

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, follow these steps:

  1. Evaluate any expressions inside parentheses.
  2. Evaluate any exponential expressions.
  3. Evaluate any multiplication and division operations from left to right.
  4. Evaluate any addition and subtraction operations from left to right.

Q: What is the difference between a variable and a constant?

A: A variable is a symbol that represents a value that can change. It is usually represented by a letter, such as x or y. A constant, on the other hand, is a value that does not change. It is usually represented by a number, such as 2 or 5.

Q: How do I substitute a value into an algebraic expression?

A: To substitute a value into an algebraic expression, replace the variable with the given value. For example, if we have the expression 2x + 3 and we want to substitute x = 4, we would replace x with 4 and get 2(4) + 3 = 8 + 3 = 11.

Q: What is the final answer to the expression 2a^2 - 2a when a = 1/4?

A: To find the final answer, we need to substitute a = 1/4 into the expression 2a^2 - 2a. This gives us 2(1/4)^2 - 2(1/4) = 2(1/16) - 2(1/4) = 1/8 - 1/2 = -3/8.

Q: Can I use a calculator or computer algebra system to check my answer?

A: Yes, you can use a calculator or computer algebra system to check your answer. This can be especially helpful if you are unsure about the final answer or if you want to verify your work.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include:

  • Failing to follow the order of operations (PEMDAS).
  • Not simplifying the expression step by step.
  • Not checking the answer using a calculator or computer algebra system.

Q: How do I know if I have simplified an algebraic expression correctly?

A: To know if you have simplified an algebraic expression correctly, you can:

  • Check your work by plugging the expression back into the original equation.
  • Use a calculator or computer algebra system to verify your answer.
  • Get feedback from a teacher or tutor.

Q: What are some real-world applications of algebraic expressions?

A: Algebraic expressions have many real-world applications, including:

  • Science: Algebraic expressions are used to model real-world phenomena, such as the motion of objects and the behavior of populations.
  • Engineering: Algebraic expressions are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Algebraic expressions are used to model economic systems and make predictions about future trends.

Conclusion

Solving algebraic expressions is a fundamental skill in mathematics, and it has many real-world applications. By following a step-by-step approach and simplifying the expression, we can arrive at the final answer. Remember to follow the order of operations (PEMDAS) and check your answer using a calculator or computer algebra system.