Problem 6 - Fundamentals of Mathematics

Question

Find the value of each expression. $$ -x^{2}+3 x-5, \text { if } x=-2. $$

Step-by-Step Solution

Answer

The value of the expression is -15.

Step 1: Substitute the Value of x

Substitute \( x = -2 \) into the expression \( -x^2 + 3x - 5 \). This step involves replacing every instance of \( x \) in the expression with \(-2\). Thus, the expression becomes:\[ -(-2)^2 + 3(-2) - 5 \].

Step 2: Simplify the Squared Term

Calculate \((-2)^2\). Squaring \(-2\) gives you:\[ (-2)^2 = 4 \].Substitute this back into the expression, which now becomes:\[ -4 + 3(-2) - 5 \].

Step 3: Simplify the Multiplication

Calculate the term \(3(-2)\). Multiplying gives:\[ 3(-2) = -6 \].Substitute this into the expression, which now reads:\[ -4 - 6 - 5 \].

Step 4: Perform Addition and Subtraction

Carefully perform the addition and subtraction from left to right:First, compute \(-4 - 6 = -10\).Then, subtract 5: \(-10 - 5 = -15\).Therefore, the expression evaluates to \(-15\).

Key Concepts

Substitution in Algebra

Substitution in algebra is a fundamental skill. It's the process of replacing variables with given values. In this exercise, we had to substitute the value of \( x = -2 \) into the algebraic expression \( -x^2 + 3x - 5 \). To do this accurately, replace every occurrence of \( x \) in the expression with \(-2\). You should be careful to use parenthesis around the number being substituted, especially when dealing with negative numbers or powers. For instance, \( (-2)^2 \) ensures that the negative stays intact until the exponent is applied. Substituting helps transform complex expressions into simpler arithmetic operations, allowing you to solve algebraic problems with ease. Here, substitution turns the expression \( -x^2 + 3x - 5 \) into a series of calculations that are easier to tackle.

Order of Operations

To evaluate any algebraic expression correctly, you must follow the order of operations. This means adhering to the sequence of calculations prescribed by PEMDAS/BODMAS:

• P/B - Parentheses/Brackets first
• E/O - Exponents/Orders (i.e., powers and square roots, etc.)
• MD/DM - Multiplication and Division (left-to-right)
• AS - Addition and Subtraction (left-to-right)

Following this order prevents calculation mistakes. In this particular problem, after substitution, we first dealt with the exponents by calculating \((-2)^2\). It's crucial to complete this step before moving to any multiplication or division operations. After that, we handled the multiplication \(3(-2)\) before finally performing addition and subtraction operations. Practicing this order ensures you can approach all algebraic problems methodically and accurately.

Negative Numbers in Algebra

Negative numbers can introduce an additional level of complexity in algebraic expressions, but understanding them can simplify the process. When substituting and manipulating negative values, careful attention to signs is critical. In this exercise, substituting a negative value, \( x = -2 \), involves several steps where negative numbers affect both squaring and multiplication. For example, \((-2)^2\) results in \(4\) because the negative sign is squared along with the number. However, \(3(-2)\) multiplies the 3 and the \(-2\) directly, resulting in \(-6\). In subtraction, such as \(-4 - 6\), think of it as a progression deeper into the negatives on a number line, resulting in \(-10\). Always double-check signs through each step. A small mistake with a negative sign can lead to an incorrect answer. By understanding how negative numbers interact with different operations, you can navigate such expressions more confidently.