Evaluate The Expression (x^2 + 3x + 3) / (x^2 - 5x - 3) When X = 3.
In the realm of mathematics, evaluating algebraic expressions is a fundamental skill. This involves substituting a given value for a variable within an expression and then performing the necessary arithmetic operations to arrive at a numerical answer. In this comprehensive guide, we will delve into the process of evaluating algebraic expressions, using the specific example of the expression (x^2 + 3x + 3) / (x^2 - 5x - 3) when x = 3. We will provide a step-by-step approach, ensuring clarity and understanding for learners of all levels.
Step 1: Understanding the Expression
Before we embark on the evaluation process, let's take a closer look at the expression itself: (x^2 + 3x + 3) / (x^2 - 5x - 3). This expression is a rational expression, which means it is a fraction where the numerator and denominator are both polynomials. The variable in this expression is 'x,' and our goal is to find the value of the expression when x is equal to 3.
Step 2: Substitution
The core of evaluating algebraic expressions lies in the act of substitution. We replace every instance of the variable 'x' in the expression with the given value, which in this case is 3. This yields the following:
((3)^2 + 3(3) + 3) / ((3)^2 - 5(3) - 3)
Step 3: Simplify the Numerator
Now that we have substituted the value of x, we can begin simplifying the expression. Let's start with the numerator: (3)^2 + 3(3) + 3. Following the order of operations (PEMDAS/BODMAS), we first evaluate the exponent:
(3)^2 = 9
Next, we perform the multiplication:
3(3) = 9
Finally, we add the terms together:
9 + 9 + 3 = 21
Therefore, the simplified numerator is 21.
Step 4: Simplify the Denominator
Now, let's turn our attention to the denominator: (3)^2 - 5(3) - 3. Again, we follow the order of operations. First, we evaluate the exponent:
(3)^2 = 9
Next, we perform the multiplication:
5(3) = 15
Now, we have:
9 - 15 - 3
Performing the subtraction from left to right, we get:
9 - 15 = -6
-6 - 3 = -9
Therefore, the simplified denominator is -9.
Step 5: Simplify the Fraction
Now that we have simplified both the numerator and the denominator, we can rewrite the expression as a fraction:
21 / -9
To further simplify this fraction, we can find the greatest common divisor (GCD) of 21 and 9, which is 3. We divide both the numerator and the denominator by 3:
21 / 3 = 7
-9 / 3 = -3
Thus, the simplified fraction is:
7 / -3
This can also be written as:
-7/3
Step 6: Final Answer
Therefore, the value of the expression (x^2 + 3x + 3) / (x^2 - 5x - 3) when x = 3 is -7/3.
Evaluating expressions is not merely a theoretical exercise; it is a practical skill that finds applications in various fields, including science, engineering, and economics. The ability to substitute values into expressions and simplify them accurately is crucial for problem-solving and decision-making. In this section, we will explore some additional tips and examples to further enhance your understanding of expression evaluation.
The Significance of Order of Operations
As we demonstrated in the previous section, the order of operations (PEMDAS/BODMAS) plays a pivotal role in evaluating expressions correctly. This mnemonic reminds us to perform operations in the following order:
- Parentheses/Brackets
- Exponents/Orders
- Multiplication and Division (from left to right)
- Addition and Subtraction (from left to right)
Failing to adhere to this order can lead to incorrect results. For instance, consider the expression 2 + 3 * 4. If we perform the addition first, we get 5 * 4 = 20, which is incorrect. The correct approach is to perform the multiplication first: 3 * 4 = 12, and then add 2: 2 + 12 = 14.
Working with Negative Numbers
When evaluating expressions, it is essential to pay close attention to the signs of the numbers involved. Negative numbers can introduce complications if not handled carefully. Remember the following rules for operations with negative numbers:
- Adding a negative number is the same as subtracting the corresponding positive number.
- Subtracting a negative number is the same as adding the corresponding positive number.
- Multiplying or dividing two numbers with the same sign (both positive or both negative) results in a positive number.
- Multiplying or dividing two numbers with different signs (one positive and one negative) results in a negative number.
Let's illustrate this with an example. Consider the expression -2x + 5 when x = -3. Substituting the value of x, we get:
-2(-3) + 5
First, we perform the multiplication:
-2 * -3 = 6
Then, we add 5:
6 + 5 = 11
Therefore, the value of the expression is 11.
Dealing with Fractions
Expressions involving fractions require careful handling to ensure accuracy. When substituting values into expressions with fractions, it is often helpful to simplify the fractions before performing other operations. This can make the calculations easier and reduce the risk of errors.
Consider the expression (1/2)x + (1/3) when x = 6. Substituting the value of x, we get:
(1/2)(6) + (1/3)
First, we perform the multiplication:
(1/2) * 6 = 3
Now we have:
3 + (1/3)
To add these terms, we need a common denominator, which is 3. We rewrite 3 as 9/3:
9/3 + 1/3
Now we can add the fractions:
(9/3) + (1/3) = 10/3
Therefore, the value of the expression is 10/3.
Utilizing Technology
In today's digital age, we have access to various tools and resources that can assist us in evaluating expressions. Calculators, both physical and online, can quickly perform arithmetic operations, including those involving exponents, fractions, and negative numbers. Additionally, computer algebra systems (CAS) such as Wolfram Alpha and Mathematica can handle more complex expressions and provide step-by-step solutions.
While technology can be a valuable aid, it is crucial to understand the underlying mathematical concepts. Relying solely on calculators without grasping the fundamentals can hinder your problem-solving abilities in the long run.
Practice Makes Perfect
Like any mathematical skill, proficiency in evaluating expressions comes with practice. The more you work through different examples and exercises, the more comfortable and confident you will become. Start with simple expressions and gradually progress to more complex ones. Don't be afraid to make mistakes; they are a natural part of the learning process. Analyze your errors and learn from them.
As you delve deeper into the world of mathematics, you will encounter more complex expressions that require advanced techniques to evaluate. These expressions may involve multiple variables, nested operations, and special functions. In this section, we will explore some of these advanced techniques and discuss how they can be applied to solve real-world problems.
Expressions with Multiple Variables
So far, we have primarily focused on expressions with a single variable. However, many real-world scenarios involve expressions with multiple variables. Evaluating such expressions requires substituting values for each variable and then performing the necessary operations.
Consider the expression 2x + 3y - z when x = 2, y = -1, and z = 4. Substituting these values, we get:
2(2) + 3(-1) - 4
First, we perform the multiplications:
4 - 3 - 4
Then, we perform the subtractions from left to right:
4 - 3 = 1
1 - 4 = -3
Therefore, the value of the expression is -3.
Nested Operations
Nested operations, also known as composite operations, involve expressions within expressions. These can be challenging to evaluate if not approached systematically. The key is to work from the innermost expressions outward.
Consider the expression 3[2(x + 1) - 5] when x = 3. Substituting the value of x, we get:
3[2(3 + 1) - 5]
First, we evaluate the innermost expression, which is the parentheses:
3 + 1 = 4
Now we have:
3[2(4) - 5]
Next, we perform the multiplication inside the brackets:
2(4) = 8
Now we have:
3[8 - 5]
Then, we perform the subtraction inside the brackets:
8 - 5 = 3
Finally, we perform the multiplication outside the brackets:
3[3] = 9
Therefore, the value of the expression is 9.
Special Functions
In advanced mathematics, you will encounter special functions such as trigonometric functions (sine, cosine, tangent), logarithmic functions, and exponential functions. Evaluating expressions involving these functions requires familiarity with their properties and values.
Consider the expression sin(x) + cos(x) when x = π/4 (where π is approximately 3.14159). Using a calculator or trigonometric table, we find that:
sin(π/4) ≈ 0.707
cos(π/4) ≈ 0.707
Therefore, the value of the expression is:
- 707 + 0.707 ≈ 1.414
Real-World Applications
Evaluating expressions is not just an academic exercise; it has numerous real-world applications. For example, in physics, we use expressions to calculate the distance traveled by an object, the force acting on an object, or the energy of a system. In finance, we use expressions to calculate interest rates, loan payments, or investment returns. In computer science, we use expressions to write algorithms and programs.
Let's consider an example from physics. The distance (d) traveled by an object moving with constant velocity (v) for a time (t) is given by the expression:
d = vt
If an object is moving with a velocity of 10 meters per second for 5 seconds, we can evaluate the expression to find the distance traveled:
d = (10 m/s)(5 s) = 50 meters
Therefore, the object travels 50 meters.
Tips for Success
To excel at evaluating expressions, keep the following tips in mind:
- Understand the order of operations (PEMDAS/BODMAS).
- Pay close attention to the signs of numbers.
- Simplify fractions before performing other operations.
- Work from the innermost expressions outward when dealing with nested operations.
- Familiarize yourself with special functions and their properties.
- Practice regularly and don't be afraid to make mistakes.
By mastering these techniques and practicing diligently, you will develop the skills necessary to confidently evaluate even the most complex expressions.
In conclusion, evaluating expressions is a fundamental skill in mathematics and a powerful tool for problem-solving in various fields. By following the step-by-step approach outlined in this guide, paying attention to the order of operations, and practicing regularly, you can master this skill and unlock its full potential. Whether you are a student, a scientist, an engineer, or simply someone who enjoys the beauty of mathematics, the ability to evaluate expressions will serve you well.
From simple arithmetic to complex calculations involving multiple variables and special functions, expression evaluation is the cornerstone of mathematical reasoning and practical application. Embrace the challenge, hone your skills, and discover the power of expression evaluation in your own endeavors.
By providing a comprehensive understanding of the concepts and techniques involved, this guide empowers you to tackle any expression evaluation problem with confidence and precision. So, go forth and explore the fascinating world of mathematics, armed with the knowledge and skills to evaluate expressions like a pro.