Problem 33 - College Algebra with Corequisite Support

Question

For the following exercises, solve for the variable. \(-(2 x)^{2}+1+3\) for \(x=2\)

Step-by-Step Solution

Answer

The result is -12.

Step 1: Substitute the Value of x

Replace the variable \( x \) with the given value 2 in the expression: \[-(2 \times 2)^{2} + 1 + 3\]

Step 2: Calculate Inside the Parentheses

First, calculate the expression inside the parentheses:\[2 \times 2 = 4\]

Step 3: Square the Result

Next, square the result from Step 2:\[4^{2} = 16\]

Step 4: Apply the Negative Sign

Apply the negative sign in front of the squared result:\[-(16) = -16\]

Step 5: Add Remaining Values

Finally, add the remaining numbers:\[-16 + 1 + 3 = -12\]

Step 6: Conclusion

The final result after substituting \( x = 2 \) into the expression is \(-12\).

Key Concepts

Step-by-step algebra solutions

When it comes to solving algebraic expressions, breaking down the problem into detailed, clear steps can help make complex equations more digestible. Step-by-step solutions are valuable because they allow students to see the logical sequence necessary to solve an expression. This systematic approach reduces confusion and makes it easier to pinpoint and correct mistakes.
Here's how step-by-step solutions work:

• Identify the problem and its requirements. Recognize the expression and the value to substitute.
• Follow through each calculation meticulously while maintaining clarity at each stage.
• Reevaluate each step to ensure the procedure is tracked correctly, leading to an accurate final solution.

This systematic breakdown reveals not only the process, but also the reasoning behind each action, allowing students to understand the underlying concepts, rather than just memorizing procedures.

Substitution method in algebra

The substitution method is a fundamental technique in algebra, primarily used to solve equations. It involves replacing variables with given numbers to simplify expressions or solve for unknown variables. This method is especially useful when we need to evaluate an algebraic expression at a specific value.
To effectively use the substitution method, follow these steps:

• Identify the variable in the expression, and note the value that will replace it.
• Carefully substitute the given value for every instance of the variable within the expression. This might involve placements within parentheses or exponents.
• Simplify according to the order of operations—start by tackling equations within parentheses and then addressing exponents, multiplication, division, addition, and subtraction.

Through substitution, complex expressions become easier to manage, allowing you to focus on simplifying the expression and achieving a solution.

Evaluating expressions at a given value

Evaluating an expression at a certain value means calculating the value of the expression after substituting the given number for the variable. It's like solving a puzzle where each piece must be addressed in order and properly.
When evaluating such expressions, it's crucial to remember:

• Always substitute the value directly and double-check your replacements to avoid errors.
• Follow the order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This helps ensure each part of the expression is dealt with correctly.
• Simplify progressively, breaking down the problem into manageable steps for easier handling and analysis.

Evaluating expressions encourages a critical understanding of how each mathematical operation impacts the overall expression, improving both skill and confidence in algebra problems.