Evaluate The Expression \[$(a+3)^2 - 2a(3-4a)\$\] When \[$a = -\frac{1}{3}\$\].
Introduction
Algebraic expressions are a fundamental concept in mathematics, and evaluating them is a crucial skill for students and professionals alike. In this article, we will evaluate the expression {(a+3)^2 - 2a(3-4a)$}$ when {a = -\frac{1}{3}$}$. We will break down the expression into smaller parts, simplify each part, and then combine them to get the final result.
Understanding the Expression
The given expression is {(a+3)^2 - 2a(3-4a)$}$. This expression involves two main operations: exponentiation and multiplication. The exponentiation operation is denoted by the superscript 2, which means that the expression inside the parentheses is squared. The multiplication operation is denoted by the multiplication symbol, which means that the two expressions are multiplied together.
Breaking Down the Expression
To evaluate the expression, we need to break it down into smaller parts. We can start by expanding the squared expression {(a+3)^2$}$.
Expanding the Squared Expression
The squared expression can be expanded using the formula {(a+b)^2 = a^2 + 2ab + b^2$}$. In this case, {a = a$}$ and {b = 3$}$. Therefore, we can expand the squared expression as follows:
{(a+3)^2 = a^2 + 2a(3) + 3^2$}$
Simplifying the expression, we get:
{(a+3)^2 = a^2 + 6a + 9$}$
Evaluating the Second Part of the Expression
The second part of the expression is {-2a(3-4a)$}$. We can simplify this expression by multiplying the two expressions together.
{-2a(3-4a) = -2a(3) + 2a(4a)$}$
Simplifying the expression, we get:
{-2a(3-4a) = -6a + 8a^2$}$
Combining the Two Parts of the Expression
Now that we have simplified the two parts of the expression, we can combine them to get the final result.
{(a+3)^2 - 2a(3-4a) = (a^2 + 6a + 9) - (-6a + 8a^2)$}$
Simplifying the expression, we get:
{(a+3)^2 - 2a(3-4a) = a^2 + 6a + 9 + 6a - 8a^2$}$
Combining like terms, we get:
{(a+3)^2 - 2a(3-4a) = -7a^2 + 12a + 9$}$
Evaluating the Expression when {a = -\frac{1}{3}$}$
Now that we have simplified the expression, we can evaluate it when {a = -\frac{1}{3}$}$. We can substitute the value of {a$}$ into the expression and simplify.
{-7a^2 + 12a + 9 = -7\left(-\frac{1}{3}\right)^2 + 12\left(-\frac{1}{3}\right) + 9$}$
Simplifying the expression, we get:
{-7a^2 + 12a + 9 = -7\left(\frac{1}{9}\right) - 4 + 9$}$
Simplifying further, we get:
{-7a^2 + 12a + 9 = -\frac{7}{9} - 4 + 9$}$
Combining the fractions, we get:
{-7a^2 + 12a + 9 = -\frac{7}{9} + \frac{45}{9}$}$
Simplifying the expression, we get:
{-7a^2 + 12a + 9 = \frac{38}{9}$}$
Therefore, the value of the expression when {a = -\frac{1}{3}$}$ is {\frac{38}{9}$}$.
Conclusion
Introduction
In our previous article, we evaluated the expression {(a+3)^2 - 2a(3-4a)$}$ when {a = -\frac{1}{3}$}$. We broke down the expression into smaller parts, simplified each part, and then combined them to get the final result. In this article, we will answer some frequently asked questions about evaluating algebraic expressions.
Q: What is an algebraic expression?
A: An algebraic expression is a mathematical expression that contains variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.
Q: How do I simplify an algebraic expression?
A: To simplify an algebraic expression, you need to follow the order of operations (PEMDAS):
- 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 evaluate an algebraic expression when given a specific value for the variable?
A: To evaluate an algebraic expression when given a specific value for the variable, you need to substitute the value of the variable into the expression and simplify.
Q: What is the difference between an algebraic expression and an equation?
A: An algebraic expression is a mathematical expression that contains variables, constants, and mathematical operations. An equation is a statement that says two expressions are equal. For example, ${2x + 3 = 5\$} is an equation, while ${2x + 3\$} is an algebraic expression.
Q: How do I solve an equation?
A: To solve an equation, you need to isolate the variable on one side of the equation. You can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
Q: What is the order of operations (PEMDAS)?
A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when evaluating an algebraic expression. The order of operations is:
- 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 evaluate an expression with multiple variables?
A: To evaluate an expression with multiple variables, you need to substitute the values of the variables into the expression and simplify. For example, if you have the expression ${2x + 3y\$} and you know that {x = 2$}$ and {y = 3$}$, you can substitute these values into the expression to get ${2(2) + 3(3) = 4 + 9 = 13\$}.
Conclusion
In this article, we answered some frequently asked questions about evaluating algebraic expressions. We covered topics such as simplifying algebraic expressions, evaluating expressions when given a specific value for the variable, and solving equations. We also discussed the order of operations (PEMDAS) and how to evaluate expressions with multiple variables. By following these tips and techniques, you can become more confident and proficient in evaluating algebraic expressions.