Problem 34 - Fundamentals of Mathematics

Question

For problems \(17-46\), find the value of each expression. $$ -h^{2}-2 h-3, \text { if } h=-4 $$

Step-by-Step Solution

Answer

The value of the expression is -11.

Step 1: Substitute the value of h

We need to evaluate the expression \(-h^{2} - 2h - 3\) for \(h = -4\). Begin by substituting \(-4\) into the expression for \(h\). The expression becomes:\[-(-4)^{2} - 2(-4) - 3\]

Step 2: Calculate the squared term

Next, calculate \((-4)^{2}\). Squaring \(-4\) gives us 16, so substitute this into the expression:\[-16 - 2(-4) - 3\]

Step 3: Calculate the linear term

The next step is to compute \(-2(-4)\). Multiplying these terms, we obtain 8. Now the expression simplifies to:\[-16 + 8 - 3\]

Step 4: Simplify the expression

Now, combine all the terms to find the result. Start with \(-16 + 8\) which is \(-8\). Then subtract 3:\[-8 - 3 = -11\]

Key Concepts

Substitution Method

The substitution method is a core concept in mathematics used for evaluating expressions. This method involves replacing variables in an expression with given numerical values to calculate the expression's final value. In this context, the expression \(-h^{2} - 2h - 3\) is evaluated by substituting \(h = -4\). Let's see why substitution is important:

• Clarification: It allows you to transform an algebraic expression into an arithmetic one that can be solved through calculation.
• Simplification: Substituting a known value reduces the complexity, converting abstract variables into concrete numbers.
• Versatility: This method applies to various types of math problems, from simple calculations to complex equations.

When using this method, always ensure that each occurrence of the variable in the expression is replaced properly. For the expression above, the substitution would look like this:\[-(-4)^{2} - 2(-4) - 3\]By following a systematic approach to substitution, you're less likely to make errors, particularly with negative signs and exponents.

Integer Operations

Integer operations are fundamental in evaluating expressions, especially when dealing with powers and multiplied terms. In our example, once the variable is substituted, we need to perform integer operations correctly.Firstly, we square the substituted integer value: since \(-4\) is the substituted value, we calculate \((-4)^{2}\):

• Squaring: \((-4)^{2} = 16\)

Squaring involves multiplying the number by itself, keeping in mind the sign. The negative sign is squared, resulting in a positive value.Next, compute the product of \(-2\) and \((-4)\):

• Multiplication: \(-2 \, \times \, -4 = 8\)

Multiplying two negative integers results in a positive product. Correctly handling these operations ensures the next steps of simplification are accurate.

Simplifying Expressions

Simplifying expressions is an essential skill in solving algebraic problems, which involves reducing the expression to its simplest form after performing operations. In the exercise's context, this process follows integer operations:Begin with the expression \[-16 + 8 - 3\]

• Combine Like Terms: Start by adding the results of the operations: \(-16 + 8 = -8\).
• Final Simplification: Subtract 3 from \(-8\), which gives \(-11\)

Simplifying means executing operations one step at a time and making sure each calculation follows the correct order. This step-by-step method helps in avoiding mistakes and ensures the final answer is correct and straightforward.