Solving (2x-3)(3x-4)-(x-73)(x-4)=40 A Step-by-Step Guide

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Hey guys! Today, we're going to dive deep into solving a fascinating equation: (2x-3)(3x-4)-(x-73)(x-4)=40. Math can seem daunting, but don't worry! We'll break it down step-by-step, making it super easy to follow. Whether you're a student tackling homework or just a math enthusiast, this guide will provide you with a clear understanding of how to solve this equation. We’ll cover everything from expanding the expressions to simplifying and finding the final solutions. So, grab your pencils, and let’s get started!

Expanding the Expressions

The first hurdle in solving this equation is expanding those expressions. When we talk about expanding the expressions, we mean getting rid of the parentheses by multiplying out the terms. This is a crucial step because it transforms the equation into a more manageable form. Let's break down each part:

Expanding (2x-3)(3x-4)

To expand (2x-3)(3x-4), we'll use the FOIL method: First, Outer, Inner, Last. This mnemonic helps us make sure we multiply every term in the first parenthesis by every term in the second.

  • First: Multiply the first terms in each parenthesis: (2x)(3x) = 6x²
  • Outer: Multiply the outer terms: (2x)(-4) = -8x
  • Inner: Multiply the inner terms: (-3)(3x) = -9x
  • Last: Multiply the last terms: (-3)(-4) = 12

Now, let's add these up: 6x² - 8x - 9x + 12. Combining the like terms (-8x and -9x), we get our first expanded expression: 6x² - 17x + 12. This is a significant milestone, guys! We’ve successfully expanded the first part of our equation.

Expanding (x-73)(x-4)

Next up, we need to expand (x-73)(x-4). We'll use the same FOIL method here, making sure we’re consistent in our approach.

  • First: Multiply the first terms: (x)(x) = x²
  • Outer: Multiply the outer terms: (x)(-4) = -4x
  • Inner: Multiply the inner terms: (-73)(x) = -73x
  • Last: Multiply the last terms: (-73)(-4) = 292

Adding these together gives us: x² - 4x - 73x + 292. Combining the like terms (-4x and -73x), we get the second expanded expression: x² - 77x + 292. Awesome! We've expanded both parts of our original equation. Remember, the key is to take it step by step and double-check your work to avoid any sneaky errors. Expanding expressions can be tricky, but with practice, it becomes second nature. Now, let’s see how these expanded forms help us simplify the entire equation.

Simplifying the Equation

Okay, we've expanded the expressions, which is fantastic progress! Now comes the crucial step of simplifying the equation. This is where we bring everything together, combine like terms, and set the stage for solving for 'x'. Trust me, this part is like putting the puzzle pieces together, and it’s super satisfying when you see it all come together.

Combining Expanded Expressions

Our original equation, (2x-3)(3x-4)-(x-73)(x-4)=40, now looks like this after expanding:

(6x² - 17x + 12) - (x² - 77x + 292) = 40

Notice the minus sign in the middle? This is super important because we need to distribute this negative sign across the second expression. Think of it like flipping the signs of everything inside the second parenthesis.

So, let’s rewrite the equation, distributing the negative sign:

6x² - 17x + 12 - x² + 77x - 292 = 40

Gathering Like Terms

Now, we’re going to gather our like terms. Like terms are those that have the same variable and exponent. In this equation, we have x² terms, x terms, and constants. Let's group them together:

  • x² terms: 6x² - x²
  • x terms: -17x + 77x
  • Constants: 12 - 292

Combining Like Terms

Next, we'll combine each group of like terms:

  • 6x² - x² = 5x²
  • -17x + 77x = 60x
  • 12 - 292 = -280

So, our equation now looks like this: 5x² + 60x - 280 = 40. See how much simpler it’s getting? We’re really making headway here, guys!

Moving the Constant to One Side

To further simplify, we want to get all our terms on one side of the equation and set it equal to zero. This is a standard step in solving quadratic equations. We’ll subtract 40 from both sides:

5x² + 60x - 280 - 40 = 0

This simplifies to:

5x² + 60x - 320 = 0

Woohoo! We've got a simplified quadratic equation. Now, we’re ready to move on to solving for 'x'. Simplifying might seem like a lot of steps, but each one brings us closer to the solution. Next, we’ll look at factoring or using the quadratic formula to find the values of 'x'.

Solving for x

Alright, we've reached the exciting part – solving for x! We've simplified our equation to 5x² + 60x - 320 = 0. Now, we need to find the values of x that make this equation true. There are a couple of ways we can tackle this: factoring and using the quadratic formula. Let's explore both.

Factoring the Quadratic Equation

Factoring involves breaking down the quadratic equation into two binomial expressions. If we can factor our equation, it's often the quickest way to find the solutions. First, let’s see if we can simplify the equation further by dividing out a common factor. Notice that 5, 60, and 320 are all divisible by 5. So, let's divide the entire equation by 5:

(5x² + 60x - 320) / 5 = 0 / 5

This gives us:

x² + 12x - 64 = 0

Now, we need to find two numbers that multiply to -64 and add up to 12. This might take a little trial and error, but let’s think it through. The pairs of factors for 64 are (1, 64), (2, 32), (4, 16), and (8, 8). We need a combination that gives us a difference of 12. Bingo! 16 and -4 work perfectly:

16 * -4 = -64

16 + (-4) = 12

So, we can rewrite our quadratic equation in factored form as:

(x + 16)(x - 4) = 0

To find the solutions, we set each factor equal to zero:

x + 16 = 0 or x - 4 = 0

Solving these gives us:

x = -16 or x = 4

Wow! We’ve found our solutions by factoring. This method is super satisfying when the numbers align perfectly. But what if factoring isn’t straightforward? That's where the quadratic formula comes to the rescue.

Using the Quadratic Formula

The quadratic formula is a universal tool for solving quadratic equations of the form ax² + bx + c = 0. The formula is:

x = [-b ± √(b² - 4ac)] / (2a)

Our simplified equation is x² + 12x - 64 = 0, so we can identify a = 1, b = 12, and c = -64. Let's plug these values into the quadratic formula:

x = [-12 ± √(12² - 4 * 1 * -64)] / (2 * 1)

Let's break it down step by step:

  1. Calculate the discriminant (the part under the square root): 12² - 4 * 1 * -64 = 144 + 256 = 400
  2. Take the square root of the discriminant: √400 = 20
  3. Plug the values back into the formula:

x = [-12 ± 20] / 2

This gives us two possible solutions:

x = (-12 + 20) / 2 = 8 / 2 = 4

x = (-12 - 20) / 2 = -32 / 2 = -16

Guess what? We got the same solutions as when we factored! This confirms that our solutions are correct. The quadratic formula is a powerful method because it works for any quadratic equation, even those that are difficult to factor. Whether you prefer factoring or the quadratic formula, the key is to practice and choose the method that feels most comfortable for you.

Verifying the Solutions

Fantastic job, guys! We've found our potential solutions: x = -16 and x = 4. But before we celebrate, it’s crucial to verify the solutions. This step ensures that our answers are correct and that we haven't made any mistakes along the way. Plugging our solutions back into the original equation helps us confirm their validity.

Substituting x = -16 into the Original Equation

Let's start by substituting x = -16 into our original equation:

(2x-3)(3x-4)-(x-73)(x-4)=40

Plugging in x = -16, we get:

(2(-16)-3)(3(-16)-4)-((-16)-73)((-16)-4)=40

Now, let's simplify step by step:

  • (2(-16)-3) = (-32 - 3) = -35
  • (3(-16)-4) = (-48 - 4) = -52
  • ((-16)-73) = -89
  • ((-16)-4) = -20

So, our equation becomes:

(-35)(-52) - (-89)(-20) = 40

Let's multiply:

  • (-35)(-52) = 1820
  • (-89)(-20) = 1780

Now, substitute these back into the equation:

1820 - 1780 = 40

40 = 40

It checks out! When x = -16, the equation holds true. This is excellent news; we’re on the right track.

Substituting x = 4 into the Original Equation

Now, let’s verify the second solution, x = 4. We’ll plug this value into our original equation:

(2x-3)(3x-4)-(x-73)(x-4)=40

Substituting x = 4, we get:

(2(4)-3)(3(4)-4)-((4)-73)((4)-4)=40

Let's simplify:

  • (2(4)-3) = (8 - 3) = 5
  • (3(4)-4) = (12 - 4) = 8
  • ((4)-73) = -69
  • ((4)-4) = 0

Our equation now looks like:

(5)(8) - (-69)(0) = 40

Multiply:

  • (5)(8) = 40
  • (-69)(0) = 0

Substitute back into the equation:

40 - 0 = 40

40 = 40

Fantastic! This solution also checks out. When x = 4, the equation is valid. Verifying our solutions is like the final stamp of approval. It gives us confidence that we’ve solved the equation correctly. We’ve shown that both x = -16 and x = 4 satisfy the original equation. Give yourselves a pat on the back – you’ve nailed it!

Conclusion

Alright guys, we did it! We successfully solved the equation (2x-3)(3x-4)-(x-73)(x-4)=40. We started by expanding the expressions, then we simplified the equation by combining like terms and setting it to zero. We found our solutions using both factoring and the quadratic formula, and finally, we verified our answers to make sure they were correct. The solutions are x = -16 and x = 4. Remember, practice makes perfect in math. The more equations you solve, the more confident you'll become. So, keep up the great work, and don't hesitate to tackle those challenging problems. You've got this!