Expressions A And B: Calculation, Expansion, Simplification

Hey everyone! Today, we're diving deep into the world of algebraic expressions, focusing on two particular examples: A = (3x + 1)(2x – 3) + (3 – 5x)(1 – x) and B = (4x – 12)(2x + 3) – 3x(1 – x) + 36. We'll tackle these expressions step-by-step, from calculating their values for a specific input to expanding and simplifying them. So, buckle up and get ready to master these algebraic maneuvers!

1) Calculating Expressions A and B for x = 1

Let's kick things off by calculating the values of our expressions A and B when x is equal to 1. This is a fundamental step in understanding how these expressions behave and sets the stage for more complex manipulations later on. To do this, we'll simply substitute '1' for every instance of 'x' in both expressions. It's like plugging in a value to see what the machine spits out!

a) Evaluating A when x = 1

Alright, let's take expression A first: A = (3x + 1)(2x – 3) + (3 – 5x)(1 – x). We're going to replace each 'x' with '1'. So, it becomes: A = (3(1) + 1)(2(1) – 3) + (3 – 5(1))(1 – 1). Now, we follow the order of operations (PEMDAS/BODMAS) – Parentheses/Brackets, Exponents/Orders, Multiplication and Division, and Addition and Subtraction. Inside the first set of parentheses, 3(1) + 1 equals 3 + 1, which is 4. Inside the second set, 2(1) – 3 equals 2 – 3, which is -1. Moving on to the next part, 5(1) is 5, so 3 – 5 is -2. Finally, 1 – 1 is 0. Putting it all together, we have A = (4)(-1) + (-2)(0). Multiplying, we get A = -4 + 0. And finally, adding, A = -4. So, when x = 1, expression A evaluates to -4. Not too shabby, right? We've successfully navigated the substitution and arithmetic to find our first value. This process is crucial for understanding how the expression behaves at specific points and serves as a building block for more advanced analysis.

b) Evaluating B when x = 1

Now, let's tackle expression B: B = (4x – 12)(2x + 3) – 3x(1 – x) + 36. Just like with expression A, we're substituting '1' for every 'x'. So, we get: B = (4(1) – 12)(2(1) + 3) – 3(1)(1 – 1) + 36. Let's break it down, following the order of operations again. Inside the first set of parentheses, 4(1) – 12 equals 4 – 12, which is -8. In the second set, 2(1) + 3 equals 2 + 3, which is 5. Moving to the next term, 3(1) is simply 3, and (1 – 1) is 0. Now we have: B = (-8)(5) – 3(0) + 36. Multiplying, we get B = -40 – 0 + 36. Finally, adding and subtracting, B = -4. So, expression B also evaluates to -4 when x = 1. This is an interesting observation! Both A and B have the same value when x = 1. But does this mean they're always the same? That's what we'll explore in the next part of our adventure. Evaluating expressions at specific points like this is a key technique in algebra, allowing us to test our understanding and make conjectures about the behavior of the expressions.

2) Can we affirm that A = B for any value of x?

Now that we know A and B have the same value when x = 1, a natural question arises: are they equal for all values of x? This is a crucial question in algebra. Just because two expressions match at one point doesn't guarantee they'll match everywhere. To answer this definitively, we need a more rigorous approach than just plugging in a single value. This is where the power of algebraic manipulation comes into play. We'll need to expand and simplify both expressions to see if they end up being identical. If they do, then we can confidently say A = B for all x. If they don't, then we know our initial observation was just a coincidence. This is the heart of algebraic proof – using the rules of algebra to transform expressions and reveal their true nature.

a) Expanding and simplifying expressions A and B

To determine if A and B are equal for all values of x, we need to expand and simplify both expressions. This process involves using the distributive property to multiply out the terms and then combining like terms to get the expressions in their simplest form. Think of it as taking apart a complex machine and then putting it back together in a more organized way. This will allow us to directly compare the structure of A and B and see if they are indeed the same.

Expanding and Reducing A

Let's start with expression A: A = (3x + 1)(2x – 3) + (3 – 5x)(1 – x). To expand the first part, (3x + 1)(2x – 3), we use the distributive property (often remembered by the acronym FOIL - First, Outer, Inner, Last). 3x multiplied by 2x gives us 6x². 3x multiplied by -3 gives us -9x. 1 multiplied by 2x gives us 2x. And 1 multiplied by -3 gives us -3. So, the first part expands to 6x² - 9x + 2x - 3. Now let's expand the second part, (3 – 5x)(1 – x). 3 multiplied by 1 is 3. 3 multiplied by -x is -3x. -5x multiplied by 1 is -5x. And -5x multiplied by -x is +5x². So, the second part expands to 3 - 3x - 5x + 5x². Now we combine these expanded forms: A = (6x² - 9x + 2x - 3) + (3 - 3x - 5x + 5x²). Next, we combine like terms. We have 6x² and 5x², which add up to 11x². For the x terms, we have -9x, +2x, -3x, and -5x. Adding these together gives us -15x. Finally, for the constant terms, we have -3 and +3, which cancel each other out. So, after expanding and simplifying, expression A becomes A = 11x² - 15x. This is a much cleaner and more manageable form of the expression, revealing its underlying quadratic structure. Simplifying algebraic expressions like this is a fundamental skill, allowing us to solve equations, analyze functions, and make further deductions about their properties.

Expanding and Reducing B

Now, let's tackle expression B: B = (4x – 12)(2x + 3) – 3x(1 – x) + 36. Again, we'll start by expanding the products. For the first part, (4x – 12)(2x + 3), we use the distributive property. 4x multiplied by 2x is 8x². 4x multiplied by 3 is 12x. -12 multiplied by 2x is -24x. And -12 multiplied by 3 is -36. So, the first part expands to 8x² + 12x - 24x - 36. Next, let's expand -3x(1 – x). -3x multiplied by 1 is -3x. And -3x multiplied by -x is +3x². So, this part expands to -3x + 3x². Now we have: B = (8x² + 12x - 24x - 36) - 3x + 3x² + 36. Let's combine like terms. We have 8x² and 3x², which add up to 11x². For the x terms, we have 12x, -24x, and -3x. Adding these together gives us -15x. For the constant terms, we have -36 and +36, which cancel each other out. So, after expanding and simplifying, expression B becomes B = 11x² - 15x. Just like with expression A, simplifying B has revealed its underlying structure. And, crucially, we can now see that A and B are identical in their simplified forms. This is the moment of truth! By systematically expanding and simplifying, we've transformed the expressions into a form where their equivalence is undeniable.

b) Can we affirm that A = B for any value of x?

After expanding and simplifying both expressions, we found that A = 11x² - 15x and B = 11x² - 15x. The moment of truth has arrived! Since both expressions simplify to the exact same form, we can confidently conclude that A = B for all values of x. This is a powerful result! We've gone from an initial observation that A and B were equal for x = 1 to a proven fact that they are equal for every possible value of x. This highlights the importance of algebraic manipulation in establishing general truths. Simply plugging in numbers can give us hints, but it's the rigorous process of expanding and simplifying that provides the definitive answer. This conclusion has significant implications. It means that any equation or problem involving A can be equivalently expressed using B, and vice versa. This can be incredibly useful in simplifying complex calculations or finding solutions to equations. This result demonstrates the power of algebraic equivalence. By manipulating expressions using the rules of algebra, we can reveal hidden relationships and establish general truths that hold for all values of the variable.

Conclusion

So, guys, we've journeyed through the world of algebraic expressions, taking a close look at A and B. We started by calculating their values for a specific input, then dove into the crucial process of expanding and simplifying. Through this, we discovered that A and B are not just equal for one value of x, but for all values of x! This adventure highlights the power of algebraic manipulation and how it allows us to uncover fundamental truths. Keep practicing these skills, and you'll be able to conquer any algebraic challenge that comes your way!