Problem 45 - Algebra 1

Question

Evaluate the expression. $$4 a+a^{2} \text { when } a=-7$$

Step-by-Step Solution

Answer

The value of the expression when \(a = -7\) is \(21\).

Step 1: Substitution of the value of 'a'

We are given that \(a = -7\). Substitute this value into the given expression \(4a + a^2\). This gives us \(4(-7) + (-7)^2\).

Step 2: Simplify the expression

Calculate the individual parts of the formula. The result is \(-28 + 49\).

Step 3: Final Calculation

The final step is to add the two numbers together, which gives the answer \(21\).

Key Concepts

Substitution

In algebra, substitution is the process of replacing a variable with a given number or expression. This is a critical step when evaluating expressions. It allows us to work with concrete numbers instead of abstract symbols.

For example, consider the expression \(4a + a^2\) and the instruction to evaluate it when \(a = -7\). Here, substitution requires you to replace every instance of \(a\) in the expression with \(-7\). This transforms the expression into \(4(-7) + (-7)^2\).

By doing this substitution, we have prepared the expression for the next step: simplification. Substitution is straightforward but essential, as doing this step incorrectly will result in the wrong final answer. Remember to carefully replace all instances of the variable with the given value.

Simplification

Simplification involves performing operations within an expression to make it easier to understand or solve. You do this by following the order of operations: Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction (PEMDAS).

In our expression, after substitution, we have \(4(-7) + (-7)^2\). Let's break it down:

• First, multiply \(4\) by \(-7\), giving \(-28\).
• Next, calculate \((-7)^2\). Since squaring a negative number yields a positive result, \((-7)^2\) becomes \(49\).

Now the expression is reduced to \(-28 + 49\). At this point, simplification has made the expression much easier to evaluate.

Understanding simplification is key, as it allows you to manipulate and solve expressions efficiently, especially when dealing with more complex expressions.

Algebraic Expressions

An algebraic expression is a mathematical phrase combining numbers, variables, and operations. These expressions can vary in complexity, ranging from simple linear expressions to complex polynomial equations.

The expression \(4a + a^2\) features both a linear component (\(4a\)) and a quadratic component (\(a^2\)). Algebraic expressions like this one serve as the building blocks of many algebra problems.

Understanding how to work with algebraic expressions involves recognizing their parts and knowing how to manipulate them using algebraic rules. This means performing operations like substitution, simplification, expanding, factoring, and rearranging equations.

Expressions are different from equations because they do not have an "equals" sign or a specified solution. Instead, they provide a framework where you can apply values or simplify to find a result. Becoming comfortable with algebraic expressions helps students progress in mathematics by learning how to systematically solve problems.