Problem 78 - Algebra 2

Question

Evaluate each expression if \(a=3, b=-2,\) and \(c=1.2 .(\text { lesson } 1-1)\) \(c^{2}-a b\)

Step-by-Step Solution

Answer

The expression evaluates to 7.44.

Step 1: Substitute Values

To solve the expression \(c^2 - ab\), we need to substitute the given values for \(a\), \(b\), and \(c\) into the expression. Here, \(a = 3\), \(b = -2\), and \(c = 1.2\). So the expression becomes \((1.2)^2 - (3)(-2)\).

Step 2: Calculate \(c^2\)

Calculate \(c^2\) by squaring the value of \(c\). So, \(1.2^2 = 1.44\).

Step 3: Calculate \(ab\)

Calculate \(ab\) by multiplying \(a\) and \(b\). Therefore, \(3 \times (-2) = -6\).

Step 4: Simplify the Expression

Now substitute and simplify the modified expression. We have \(1.44 - (-6)\), which simplifies to \(1.44 + 6 = 7.44\).

Key Concepts

Evaluation of Expressions

When we talk about evaluation, we mean the process of finding the numerical value of an expression. An algebraic expression can involve variables, numbers, and arithmetic operations like addition and multiplication.
To evaluate an expression:

• First, substitute the given values for each variable in the expression.
• Follow the order of operations to simplify the expression and find its value.

For instance, in the expression \(c^2 - ab\), we're evaluating it for specific values of \(a, b,\) and \(c\). Each variable's value is substituted to transform the algebraic expression into a number, so long as we adhere to the fundamental mathematical principles like parentheses, exponents, and so forth. This structured substitution and simplification lead to the final answer of \(7.44\) in the example given.

Substitution in Expressions

Substitution involves replacing variables with specific values. This is a crucial step in evaluating expressions because it breaks down an abstract formula into a concrete numerical problem.
When substituting:

• Replace each variable with its given numerical value.
• Punctuation is key; keep track of negative and positive values.
• Maintain consistency in the replacement to avoid confusion.

In our example, the variables \(a, b,\) and \(c\) are given the values 3, -2, and 1.2, respectively. Thus, the original expression \(c^2 - ab\) becomes \((1.2)^2 - (3)(-2)\). Careful substitution makes further steps easier to execute and helps in avoiding small errors that can lead to incorrect results.

Arithmetic Operations

Arithmetic operations are fundamental in simplifying any algebraic expression once substitution has occurred. Involvement of basic operations such as addition, subtraction, multiplication, and exponentiation are typical.
Key points for arithmetic operations:

• Follow the correct order: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right) referred to as PEMDAS.
• Handle negative numbers with care, as subtraction of a negative is equivalent to addition.

In the given solution, two arithmetic steps were vital:

• Squaring the decimal \(c\) to find \(c^2\) as \(1.2^2 = 1.44\).
• Multiplying \(a\) and \(b\) to find \(ab\), with \(3 \times (-2) = -6\).

Finally, handling the subtraction of a negative number becomes addition, converting \(1.44 - (-6)\) into \(1.44 + 6\), resulting in \(7.44\). This illustrates how foundational arithmetic techniques are in reaching the desired solution.