Problem 10 - Algebra 1

Question

Evaluate the expression when \(x=3\) $$ 2 x^{2} $$

Step-by-Step Solution

Answer

The value of the expression \(2 x^{2}\) when \(x=3\) is \(18\)

Step 1: Substitute the given value for x

Replace \(x\) in the expression \(2 x^{2}\) with the given value of \(3\). This results in the expression being \(2*3^{2}\)

Step 2: Simplify the expression

Now, simplify the expression. The expression \(3^{2}\) equals to \(9\), and then \(2*9\) equals to \(18\). So, \(2*3^{2}\) simplifies to \(18\)

Key Concepts

Substitution Method in Algebra

Understanding the substitution method is essential for evaluating expressions in algebra. This method involves replacing a variable in an expression with a given numerical value. For instance, if you have the expression 2x^2 and you are told that x=3, the first step is to substitute 3 for each occurrence of x in the expression. After the substitution, you would have 2*3^2, setting a clear pathway to simplifying the expression further.

It is crucial to follow the proper order of operations after substitution – first calculating exponents, then multiplication or division, and finally addition or subtraction. This ensures that you accurately evaluate the given expression. Remember, correctly substituting values is much like putting the correct key into a lock: it allows the rest of the process to turn smoothly, leading to the 'unlocking' of the expression's value.

Simplifying Expressions

Simplifying expressions is a process that involves reducing an algebraic expression to its simplest form. In our example, after substituting 3 for x, we end up with 2*3^2. Simplifying this expression should be done in compliance with the rules of arithmetic operations. According to the order of operations, we first resolve the exponent (3^2), which means multiplying 3 by itself to get 9. Next, we perform the multiplication with the remaining part of the expression, which is 2*9. Thus, the fully simplified form of the expression is 18.

When simplifying expressions, it's important to work methodically to avoid errors. Careful attention to detail ensures each step is correctly executed, which is particularly imperative when dealing with more complex expressions that include a mixture of operations.

Working with Exponents

Exponents are a foundational concept in algebra that indicate the power to which a number is raised. They form an integral part of evaluating algebraic expressions. In the expression 2x^2, the exponent is 2, which tells us to multiply x by itself. When you are given a value for x, like 3 in our exercise, you would calculate 3^2 which is 3*3 and equals 9.

Remember, an exponent applies only to the number it is directly attached to unless parentheses indicate a broader application. For example, 2x^2 is different from (2x)^2. The latter means you would square the entire term 2x, which includes multiplying 2 by itself as well as x by itself, leading to a different result than if you were to square only x and then multiply by 2. Misinterpreting the use of exponents can significantly alter the outcome of the expression, so it is pivotal to understand and apply the exponent rules correctly.