Solving A = (2x-3)(-5+3x)-4+6x: A Math Guide
Hey guys! Let's dive into solving this equation step by step. This guide will break down the process, making it super easy to understand. We'll cover everything from the initial setup to the final solution. So, if you're scratching your head wondering how to tackle this, you're in the right place! Whether you're a student prepping for an exam or just someone who loves math, this breakdown will help.
Understanding the Problem
Before we jump into solving, let’s make sure we understand the equation. The equation we're dealing with is A = (2x-3)(-5+3x)-4+6x. Our goal here is to simplify this expression and, if possible, solve for x. This involves expanding the product of the binomials, combining like terms, and potentially solving a quadratic equation. It might sound like a mouthful, but trust me, we'll take it one step at a time.
The first key step in solving this equation is to recognize the structure. We have a product of two binomials (2x-3) and (-5+3x), which we'll need to expand using the distributive property (also known as the FOIL method). We also have some additional terms (-4 and +6x) that we’ll need to combine with the expanded terms. Recognizing this structure helps us to plan our approach and avoid common mistakes.
Why is understanding the structure so important? Well, it’s like having a roadmap before starting a journey. Without it, you might take wrong turns or get lost. In math, understanding the structure of an equation helps you choose the right operations and in the correct order. This prevents you from making algebraic errors and ensures you’re on the right track to the solution. So, let’s keep this roadmap in mind as we proceed!
Step 1: Expanding the Binomials
The first major step in solving our equation is to expand the product of the two binomials: (2x - 3)(-5 + 3x). We'll use the distributive property (often remembered by the acronym FOIL – First, Outer, Inner, Last) to multiply each term in the first binomial by each term in the second binomial. This is a crucial step, so let's break it down slowly and methodically.
So, what does FOIL mean exactly? It stands for:
- First: Multiply the first terms in each binomial: (2x) * (3x)
- Outer: Multiply the outer terms in the binomials: (2x) * (-5)
- Inner: Multiply the inner terms in the binomials: (-3) * (3x)
- Last: Multiply the last terms in each binomial: (-3) * (-5)
Following these steps, let’s do the multiplication:
- First: (2x) * (3x) = 6x²
- Outer: (2x) * (-5) = -10x
- Inner: (-3) * (3x) = -9x
- Last: (-3) * (-5) = 15
Now we add these products together: 6x² - 10x - 9x + 15. This gives us the expanded form of the binomial product. Next, we'll combine the like terms to simplify this expression further. Expanding binomials might seem daunting at first, but with practice, it becomes second nature. Keep in mind the FOIL method, and you'll be expanding binomials like a pro in no time!
Step 2: Combining Like Terms
After expanding the binomials, we have the expression 6x² - 10x - 9x + 15. Now, we need to combine the like terms to simplify this expression. Like terms are terms that have the same variable raised to the same power. In this case, -10x and -9x are like terms because they both have the variable x raised to the power of 1. Let's see how we can simplify this further.
To combine like terms, we simply add or subtract their coefficients. The coefficient is the number that multiplies the variable. So, for -10x and -9x, the coefficients are -10 and -9, respectively. Adding these coefficients gives us: -10 + (-9) = -19. Therefore, -10x - 9x combines to -19x. Now we can rewrite our expression as: 6x² - 19x + 15.
This simplified form is much easier to work with. We've reduced the number of terms and made the expression more manageable. Combining like terms is a fundamental skill in algebra, and it’s essential for simplifying expressions and solving equations. It’s like decluttering a room – once you tidy up, everything becomes much clearer and easier to handle. So always remember to look for like terms and combine them to make your algebraic expressions cleaner and more straightforward!
Step 3: Incorporating the Remaining Terms
Now that we've expanded and simplified the binomial part of our equation, let’s bring in the remaining terms. Our original equation was A = (2x - 3)(-5 + 3x) - 4 + 6x. We've already simplified (2x - 3)(-5 + 3x) to 6x² - 19x + 15. Now we need to add the -4 and +6x terms into the mix. This is where we complete the simplification process by combining all like terms in the full equation.
So, let’s rewrite the full equation with our simplified binomial expression:
A = 6x² - 19x + 15 - 4 + 6x
Now we look for like terms again. We have -19x and +6x, which are like terms because they both have x raised to the power of 1. We also have the constants +15 and -4, which are also like terms. Let’s combine these:
- Combining the x terms: -19x + 6x = -13x
- Combining the constants: 15 - 4 = 11
After combining these like terms, our equation simplifies to:
A = 6x² - 13x + 11
This is the simplified form of the original equation. We've expanded the binomials, combined like terms, and now have a quadratic expression. At this point, the equation is much cleaner and easier to analyze. Incorporating the remaining terms and simplifying is like putting the final touches on a painting – it brings all the elements together to create a complete picture.
Step 4: Analyzing the Simplified Equation
After all the simplification, we’ve arrived at the equation A = 6x² - 13x + 11. Now, let's take a moment to analyze what we have. This equation is a quadratic equation in the form of Ax² + Bx + C, where A = 6, B = -13, and C = 11. Understanding the type of equation we’re dealing with is crucial because it dictates the methods we can use to solve it. In this case, we have a couple of options: we can either try to factor the quadratic, use the quadratic formula, or complete the square.
Why is analyzing the simplified equation so important? It's like taking a moment to read the map before deciding on the best route. By recognizing the form of the equation, we can choose the most efficient method to solve for x. For example, if the quadratic factors easily, that’s often the quickest route. If not, the quadratic formula provides a reliable way to find the solutions. Completing the square is another valid method, particularly useful in certain contexts like deriving the quadratic formula itself.
So, before jumping into the solution, let’s consider our options. We’ll briefly explore whether the quadratic can be factored easily, and then we might lean towards using the quadratic formula for a straightforward solution. Remember, the goal is not just to get an answer, but to understand the process and why we choose certain methods. Analyzing the equation is a key part of this understanding, helping us to develop a strategic approach to problem-solving.
Step 5: Attempting to Factor the Quadratic (Optional)
Before we jump straight to the quadratic formula, let’s explore the possibility of factoring our quadratic equation, 6x² - 13x + 11. Factoring, if possible, can be a quicker way to find the solutions. However, not all quadratics are easily factorable, and it’s important to recognize when factoring might not be the most efficient approach. So, let’s give it a try and see if we can break this quadratic down into simpler terms.
To factor a quadratic expression in the form of Ax² + Bx + C, we need to find two numbers that multiply to AC and add up to B. In our case, A = 6, B = -13, and C = 11. So, we’re looking for two numbers that multiply to (6)(11) = 66 and add up to -13. This is where the trial and error comes in.
Let's list some factor pairs of 66:
- 1 and 66
- 2 and 33
- 3 and 22
- 6 and 11
Now, we need to see if any of these pairs (or their negative counterparts) add up to -13. After reviewing these pairs, it becomes clear that none of them add up to -13. This suggests that our quadratic equation 6x² - 13x + 11 may not be easily factorable using simple integers. This is valuable information because it saves us time from trying more complex factoring methods that might not lead to a solution.
Recognizing that a quadratic might not be easily factorable is a crucial skill. It allows us to pivot to other methods, like the quadratic formula, without wasting time on a fruitless endeavor. Factoring is a powerful tool, but it’s not always the best tool for the job. So, with this in mind, let’s move on to a more reliable method for solving quadratic equations: the quadratic formula.
Step 6: Applying the Quadratic Formula
Since we determined that factoring our quadratic equation 6x² - 13x + 11 might not be straightforward, let's use the quadratic formula. The quadratic formula is a reliable method for finding the solutions (or roots) of any quadratic equation in the form Ax² + Bx + C = 0. It might look a bit intimidating at first, but once you understand how to use it, it’s a powerful tool in your math arsenal. So, let's break it down step by step.
The quadratic formula is given by:
x = [ -B ± √(B² - 4AC) ] / (2A)
Where A, B, and C are the coefficients from our quadratic equation. In our case, we have A = 6, B = -13, and C = 11. Now, we simply plug these values into the formula and simplify. Let’s start by substituting the values:
x = [ -(-13) ± √((-13)² - 4(6)(11)) ] / (2(6))
Next, we simplify the expression inside the square root:
- (-13)² = 169
- 4(6)(11) = 264
- 169 - 264 = -95
So our equation now looks like this:
x = [ 13 ± √(-95) ] / 12
Notice that we have a negative number inside the square root. This means that the solutions will be complex numbers. Complex numbers involve the imaginary unit i, where i² = -1. So, we can rewrite √(-95) as √(95) * √(-1), which is √(95) * i. Our equation now becomes:
x = [ 13 ± i√(95) ] / 12
We can express the solutions as two separate complex numbers:
- x₁ = (13 + i√(95)) / 12
- x₂ = (13 - i√(95)) / 12
These are the solutions to our quadratic equation. The quadratic formula is a powerful tool because it guarantees a solution, whether the roots are real or complex. So, even when factoring seems difficult or impossible, the quadratic formula provides a reliable path to the answer. Keep this formula handy, and you’ll be able to tackle any quadratic equation that comes your way!
Step 7: Interpreting the Solutions
We’ve arrived at the solutions for our equation A = 6x² - 13x + 11, which are complex numbers:
- x₁ = (13 + i√(95)) / 12
- x₂ = (13 - i√(95)) / 12
Now, let's take a moment to interpret what these solutions mean. In the context of quadratic equations, the solutions (also called roots or zeros) are the values of x that make the equation equal to zero. However, in our original problem, we were given A = 6x² - 13x + 11, and we haven't set A to any specific value, such as zero. So, these solutions are the values of x for which A would equal zero if we had the equation 0 = 6x² - 13x + 11.
Since our solutions are complex numbers, this tells us that there are no real number values of x that will make the expression 6x² - 13x + 11 equal to zero. Graphically, this means that the parabola represented by the quadratic equation does not intersect the x-axis. The curve either lies entirely above or entirely below the x-axis.
If we were trying to solve a real-world problem, complex solutions might indicate that there is no real-world answer. For example, if we were modeling the trajectory of a ball, and we got complex solutions when trying to find when the ball hits the ground, it would mean that, according to our model, the ball never hits the ground. In pure mathematics, complex solutions are perfectly valid and provide a complete picture of the solution space.
Interpreting the solutions is just as crucial as finding them. It allows us to understand the implications of our results and whether they make sense in the context of the problem. Whether you're dealing with real or complex numbers, knowing how to interpret your solutions will help you to apply math effectively in various situations.
Conclusion
Alright guys, we've reached the end of our journey! We successfully solved the equation A = (2x - 3)(-5 + 3x) - 4 + 6x. We expanded the binomials, combined like terms, and ended up with a quadratic equation. We then used the quadratic formula to find the solutions, which turned out to be complex numbers.
Let's recap the key steps we took:
- Expanded the binomials: Using the FOIL method, we multiplied (2x - 3) and (-5 + 3x).
- Combined like terms: We simplified the expression by adding and subtracting terms with the same variable and power.
- Incorporated remaining terms: We included the -4 and +6x from the original equation and combined like terms again.
- Analyzed the simplified equation: We recognized that we had a quadratic equation and considered our options for solving it.
- Attempted to factor (optional): We tried factoring, but determined it wasn't straightforward.
- Applied the quadratic formula: We used the quadratic formula to find the solutions, which turned out to be complex numbers.
- Interpreted the solutions: We understood that complex solutions mean there are no real number solutions for when the expression equals zero.
Solving this equation involved several algebraic techniques, and each step was crucial to arriving at the final answer. Remember, math isn't just about getting the right answer; it’s about understanding the process. By breaking down the problem into manageable steps, we made it easier to tackle. So next time you encounter a similar problem, remember the steps we’ve discussed, and you’ll be well-equipped to solve it! Keep practicing, and you'll become a math whiz in no time!