Evaluating 2x² + 5x - 3 At X = 2 A Step-by-Step Guide
Hey there, math enthusiasts! Ever stumbled upon an algebraic expression and felt a twinge of curiosity about its value when a specific number steps in for the variable? Well, today, we're diving headfirst into just that! We're going to explore the ins and outs of evaluating the expression 2x² + 5x - 3 when x takes on the value of 2. Buckle up, because this journey will not only demystify this particular problem but also equip you with the skills to tackle similar challenges with confidence. Let's get started, shall we?
Decoding the Expression: 2x² + 5x - 3
Before we jump into the calculation, let's break down the expression 2x² + 5x - 3. Think of it as a mathematical puzzle, with each piece playing a crucial role. The expression comprises three terms: 2x², 5x, and -3. The first term, 2x², involves squaring the variable x and then multiplying the result by 2. The second term, 5x, is simply 5 times x. And finally, we have the constant term, -3, which stands alone without any variable attached. Understanding these components is key to navigating the evaluation process smoothly. This is basically an introduction to polynomial expressions. A polynomial expression is a mathematical expression involving variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents. The expression 2x² + 5x - 3 perfectly fits this definition, making it a classic example of a polynomial expression. Polynomial expressions come in various forms, from simple linear expressions like 3x + 2 to more complex forms like 4x³ - 2x² + x - 7. Each term in a polynomial expression consists of a coefficient (the numerical factor) and a variable raised to a non-negative integer power. The degree of a term is the power to which the variable is raised, and the degree of the polynomial is the highest degree among all its terms. For instance, in the expression 2x² + 5x - 3, the term 2x² has a degree of 2, the term 5x has a degree of 1 (since x is implicitly raised to the power of 1), and the term -3 has a degree of 0 (since it can be seen as -3x⁰). Therefore, the degree of the polynomial expression is 2, which makes it a quadratic expression. Understanding these fundamental concepts about polynomial expressions will not only help you solve problems like the one we're tackling today but also lay a solid foundation for more advanced algebraic topics. So, keep these concepts in mind as we move forward in our mathematical journey!
The Substitution Step: Plugging in x = 2
Now comes the exciting part – substituting the value of x with 2! This is where the magic happens, guys. We're essentially replacing the unknown x with a concrete number, paving the way to a numerical answer. In the expression 2x² + 5x - 3, wherever you see an x, you'll replace it with 2. So, 2x² becomes 2(2)², 5x transforms into 5(2), and the constant term -3 remains untouched. Remember, the order of operations is crucial here. We need to handle the exponent first, then multiplication, and finally addition and subtraction. This methodical approach ensures we arrive at the correct value without any hiccups. Imagine trying to build a house without a blueprint – things could get messy, right? Similarly, in mathematics, the order of operations acts as our blueprint, guiding us through complex calculations with clarity and precision. The acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) is a handy tool for remembering this order. It's like a mathematical GPS, keeping us on the right track. So, when we substitute x with 2 in the expression 2x² + 5x - 3, we're setting the stage for a series of operations that must be performed in the correct sequence. We first encounter 2(2)², which involves both an exponent and multiplication. According to PEMDAS, exponents take precedence, so we must square 2 before multiplying by 2. Then we have 5(2), which is a straightforward multiplication. Finally, we have the constant term -3, which remains unchanged. By meticulously following the order of operations, we ensure that we're performing each step in the correct sequence, preventing any errors along the way. This careful approach not only helps us solve this specific problem accurately but also instills a valuable habit of precision and attention to detail, which is crucial for success in mathematics and beyond.
Evaluating the Expression: A Step-by-Step Guide
Let's dive into the actual evaluation! After substituting x = 2, our expression looks like this: 2(2)² + 5(2) - 3. Now, we follow the order of operations, starting with the exponent. 2² is 4, so we have 2(4) + 5(2) - 3. Next up is multiplication: 2(4) = 8 and 5(2) = 10, giving us 8 + 10 - 3. Finally, we perform addition and subtraction from left to right: 8 + 10 = 18, and 18 - 3 = 15. So, the value of the expression 2x² + 5x - 3 when x = 2 is 15. Yay, we cracked it! Think of it like solving a jigsaw puzzle – each piece (or operation) fits perfectly into place, leading us to the final picture (or the solution). And just like a well-executed puzzle, our step-by-step evaluation ensures accuracy and clarity. Let's delve deeper into each step to solidify our understanding. First, we tackled the exponent, 2², which is simply 2 multiplied by itself, resulting in 4. This step is crucial because exponents represent repeated multiplication, and handling them correctly is paramount. Next, we moved on to the multiplications: 2(4) and 5(2). Remember, multiplication is a shortcut for repeated addition, and in these cases, we're essentially adding 4 to itself twice and 2 to itself five times. Performing these multiplications accurately is vital for obtaining the correct final answer. Finally, we arrived at the addition and subtraction: 8 + 10 - 3. Addition and subtraction are inverse operations, and we perform them from left to right, just like reading a sentence. So, we first added 8 and 10 to get 18, and then subtracted 3 from 18 to arrive at our final answer of 15. Each step in this evaluation process is a building block, contributing to the overall solution. By understanding the underlying principles and carefully executing each operation, we not only solve this particular problem but also reinforce our mathematical skills and confidence. So, let's celebrate our accomplishment and carry this newfound knowledge forward!
Why This Matters: Applications in Real Life
You might be wondering,