Problem 52 - Prealgebra and Introductory Algebra

Question

Evaluate the variable expression for \(a=-2, b=4, c=-1,\) and \(d=3\) $$(b+d)^{2}-4 a$$

Step-by-Step Solution

Answer

The expression \((b+d)^{2}-4 a\) evaluates to 57 when \(a=-2, b=4, c=-1,\) and \(d=3\).

Step 1: Substituting variables.

First, substitute the given values into the expression. This gives us the following: \[ \((4+3)^{2}-4*(-2)\]

Step 2: Applying Order of Operations Part 1 - Calculating the Parenthetical Value.

Then, calculate within the parentheses: \[ \((7)^{2}-4*(-2)\]

Step 3: Applying Order of Operations Part 2 - Exponentiation.

Next, take the square of 7 (raise 7 to the power of 2) and keep the rest as it is: \[ \(49-4*(-2)\]

Step 4: Applying Order of Operations Part 3 - Multiplication.

Multiply -4 and -2: \[ \(49+8)\]

Step 5: Applying Order of Operations Part 4 - Subtraction.

Finally, add the two resulting numbers together to reach the solution: \[ \(57)\]

Key Concepts

Understanding the Order of Operations

When tackling any mathematical expression, it's crucial to follow the order of operations, often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This sequence ensures that we solve expressions accurately step by step.

In our exercise, we used this order to evaluate \( (b + d)^2 - 4a \):

• First, handle operations inside parentheses: \( (4 + 3) \).
• Next, compute exponents: square the result from the parentheses.
• Then, perform multiplication.
• Finally, conclude with subtraction or addition.

This logical order prevents errors and helps us reach the correct solution seamlessly.

Substitution in Expressions: Simplifying Through Replacement

Substitution is a key mathematical technique where we replace variables in an expression with given numerical values. This step transforms a variable expression into a numerical one, making it easier to solve.

In this exercise, substituting \( a = -2 \), \( b = 4 \), \( c = -1 \), and \( d = 3 \) into \( (b + d)^2 - 4a \) gives us:

• Replace \( b \) with 4 and \( d \) with 3 in the parentheses.
• Substitute \( a \) with -2 in front of the multiplication.

Now, the expression becomes purely numerical: \( (4 + 3)^2 - 4(-2) \). This simplification helps focus purely on computation, following the order of operations.

Enhancing Mathematical Problem-Solving Skills

Developing strong problem-solving skills in math is about understanding concepts deeply and applying them accurately. Successfully solving problems requires a clear process.

Here's a basic approach to enhance these skills:

• Carefully read and identify what's being asked in the problem.
• Substitute known values into variable expressions without rushing.
• Apply the order of operations methodically.
• Double-check each step to prevent errors.

By consistently practicing these techniques, students can build confidence and improve their ability to tackle a wide range of mathematical challenges.