Problem 62 - Algebra 1: Concepts and Skills

Question

Evaluate the expression \(2 x^{2}\) when \(x=5\) (F) 10 (G) 100 (H) 1000 (J) 10000

Step-by-Step Solution

Answer

None of the provided options is correct. The evaluated expression \(2 x^{2}\) where \(x=5\) equals \(50\).

Step 1: Substitute the value of x

Replace \(x\) in the expression \(2 x^{2}\) with the given value \(5\). The expression then becomes \(2 \cdot 5^{2}\).

Step 2: Evaluate the Power

Now, calculate the value \(5^{2}\). It is crucial to evaluate any powers before proceeding with other operations. The expression becomes \(2 \cdot 25\).

Step 3: Perform Multiplication

Finally, multiply \(2\) by \(25\). The result is \(50\).

Key Concepts

Exponentiation

Exponentiation is a fundamental mathematical operation that involves raising a number to the power of an exponent. In the given exercise, the expression contains the term \(x^{2}\). The number \(x\) is called the base, and the small number \(2\) that is written above the base is the exponent. The operation instructs us to multiply the base by itself.
This means that \(x^{2}\) is the same as \(x \times x\). Using exponentiation, if the base \(x\) is given as \(5\), as in this case, the expression \(5^{2}\) translates to \(5 \times 5\), which equals \(25\).
Understanding exponentiation is vital for solving more complex algebraic expressions, as it simplifies repeated multiplication into a compact form.

Substitution

Substitution is the process of replacing a variable in an algebraic expression with a given value. For this problem, we started with the expression \(2x^{2}\), where \(x\) is the variable. The exercise specifies that \(x = 5\).
To perform substitution, we replace every occurrence of \(x\) in the expression with the value \(5\). This transforms our expression to \(2 \cdot 5^{2}\). Substitution is essential in evaluating algebraic expressions because it allows for the transformation of a general expression into something that can be calculated directly.
Through substitution, we can turn abstract algebraic problems into concrete numerical solutions.

Order of Operations

The order of operations is a rule used to clarify which procedures to perform first in a given mathematical expression. It's key when evaluating complex expressions like in the given exercise. The general order is often remembered through the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
In our problem, after substitution, we have the expression \(2 \cdot 5^{2}\). According to the order of operations, exponentiation is evaluated before multiplication. This means that \(5^{2}\) (the exponentiation) must be computed first, giving us \(25\). Only after this step do we proceed to multiplication, where \(2 \cdot 25\) is calculated as \(50\).
Following the correct order of operations ensures that our final solution is accurate and consistent.

Evaluation of Expressions

Evaluation of expressions involves finding the numerical value of an algebraic expression by applying operations such as substitution and following the order of operations. The key here is to reduce the expression step by step until you arrive at a single numerical result.
After substituting \(x = 5\) into our expression and performing the exponentiation as per the order of operations, we are left with \(2 \cdot 25\). From here, it’s straightforward to complete the multiplication: \(2 \times 25 = 50\).
Evaluation is an integral part of solving algebraic problems as it transitions abstract mathematical concepts into practical, usable information. This process of systematic reduction ensures clarity and correctness in finding the solution.