Solve (2x - 4)² = (7x + 17)²: Find X Values

by Mireille Lambert 44 views

Hey everyone! Today, we're diving into a really interesting math problem: solving the equation (2x - 4)² = (7x + 17)². This kind of problem might seem intimidating at first, but don't worry, we'll break it down step by step. We'll explore the different methods to find the values of x, discuss the possible solutions, and, most importantly, learn how to verify if our answers are correct. So, grab your calculators and let's get started!

Understanding the Problem

Before we jump into the solution, let's make sure we fully understand what we're dealing with. The equation (2x - 4)² = (7x + 17)² involves squared binomials. This means we're essentially dealing with a quadratic equation in disguise. Our goal is to find the values of x that make this equation true. There are a couple of ways we can approach this, and we'll explore both to give you a solid understanding.

First, let’s think about what squaring something means. When we square a number or an expression, we're multiplying it by itself. So, (2x - 4)² means (2x - 4) * (2x - 4), and similarly, (7x + 17)² means (7x + 17) * (7x + 17). Knowing this is crucial because it helps us expand these expressions correctly.

Next, consider the structure of the equation. We have two squared expressions set equal to each other. This opens up a couple of avenues for solving it. We could expand both sides, simplify, and rearrange the equation into a standard quadratic form (ax² + bx + c = 0). Alternatively, we can use the property that if a² = b², then a = b or a = -b. This property can save us some steps and potential algebraic headaches. We’ll delve into both methods shortly!

Understanding the core concepts – what squaring means and the different approaches we can take – is the first step to conquering this problem. So, let’s move on and explore those solution methods.

Method 1: Expanding and Simplifying

The first method we'll explore involves expanding both sides of the equation, simplifying, and then rearranging it into the standard quadratic form. This method is a classic approach to solving equations like this and provides a solid foundation for understanding quadratic equations in general. Let’s walk through the steps together.

Step 1: Expand Both Sides

Remember, (2x - 4)² means (2x - 4) * (2x - 4), and (7x + 17)² means (7x + 17) * (7x + 17). We'll use the FOIL method (First, Outer, Inner, Last) or the distributive property to expand these binomials. Guys, it’s crucial to be careful with your signs and multiplication here – a small mistake can throw off the whole solution!

  • Expanding (2x - 4)²:

    • (2x - 4) * (2x - 4) = (2x * 2x) + (2x * -4) + (-4 * 2x) + (-4 * -4)
    • = 4x² - 8x - 8x + 16
    • = 4x² - 16x + 16

  • Expanding (7x + 17)²:

    • (7x + 17) * (7x + 17) = (7x * 7x) + (7x * 17) + (17 * 7x) + (17 * 17)
    • = 49x² + 119x + 119x + 289
    • = 49x² + 238x + 289

Step 2: Set Up the Equation

Now that we've expanded both sides, our equation looks like this:

  • 4x² - 16x + 16 = 49x² + 238x + 289

Step 3: Rearrange into Standard Quadratic Form

To solve a quadratic equation, we want it in the form ax² + bx + c = 0. This means we need to move all the terms to one side of the equation. Let’s subtract the left side from the right side to keep the x² term positive (this just makes the factoring or quadratic formula steps a bit easier).

  • 0 = 49x² + 238x + 289 - (4x² - 16x + 16)
  • 0 = 49x² + 238x + 289 - 4x² + 16x - 16
  • 0 = 45x² + 254x + 273

Step 4: Solve the Quadratic Equation

Now we have a standard quadratic equation: 45x² + 254x + 273 = 0. There are a few ways to solve this:

  1. Factoring: We try to factor the quadratic expression into two binomials. This can be tricky, especially when the coefficients are large.
  2. Quadratic Formula: The quadratic formula is a reliable method that always works. It states that for an equation ax² + bx + c = 0, the solutions for x are given by:
    • x = (-b ± √(b² - 4ac)) / (2a)

Let's use the quadratic formula here. In our case, a = 45, b = 254, and c = 273.

  • x = (-254 ± √(254² - 4 * 45 * 273)) / (2 * 45)
  • x = (-254 ± √(64516 - 49140)) / 90
  • x = (-254 ± √15376) / 90
  • x = (-254 ± 124) / 90

This gives us two possible solutions:

  • x₁ = (-254 + 124) / 90 = -130 / 90 = -13 / 9
  • x₂ = (-254 - 124) / 90 = -378 / 90 = -21 / 5

So, using the expanding and simplifying method, we found two potential solutions: x = -13/9 and x = -21/5. But remember, we always need to verify our solutions to make sure they work in the original equation! We’ll get to verification later.

Method 2: Using the Square Root Property

Okay, guys, now let's explore another method to solve the equation (2x - 4)² = (7x + 17)². This method uses the square root property, which can be a more direct approach in certain situations. Remember, the square root property states that if a² = b², then a = b or a = -b. This is a neat trick that can save us from having to expand and rearrange the entire equation.

Step 1: Apply the Square Root Property

Since we have two squared expressions equal to each other, we can directly apply the square root property. This gives us two separate equations:

  1. 2x - 4 = 7x + 17
  2. 2x - 4 = -(7x + 17)

Notice how we’ve created two linear equations from our original squared equation. This is a significant simplification!

Step 2: Solve the First Equation

Let's solve the first equation: 2x - 4 = 7x + 17. Our goal is to isolate x. Let's start by moving the x terms to one side and the constants to the other.

  • Subtract 2x from both sides: -4 = 5x + 17
  • Subtract 17 from both sides: -21 = 5x
  • Divide by 5: x = -21/5

So, one potential solution is x = -21/5. This is one of the solutions we found using the expanding and simplifying method. That’s a good sign!

Step 3: Solve the Second Equation

Now let’s solve the second equation: 2x - 4 = -(7x + 17). Again, we want to isolate x, but first, we need to distribute the negative sign on the right side.

  • 2x - 4 = -7x - 17
  • Add 7x to both sides: 9x - 4 = -17
  • Add 4 to both sides: 9x = -13
  • Divide by 9: x = -13/9

Our second potential solution is x = -13/9. This matches the other solution we found earlier! Isn't it cool how different methods can lead us to the same answers?

So, using the square root property method, we also found two potential solutions: x = -21/5 and x = -13/9. Now, before we declare victory, we have one crucial step left: verifying these solutions.

Verifying the Solutions

Okay, we've found two potential solutions for the equation (2x - 4)² = (7x + 17)²: x = -13/9 and x = -21/5. But here's a super important step in math: always verify your solutions! Plugging our answers back into the original equation ensures that they actually work and that we haven't made any mistakes along the way. It’s like double-checking your work before submitting a test – you want to be sure you’ve got it right!

Step 1: Verify x = -13/9

Let's substitute x = -13/9 into the original equation:

  • (2(-13/9) - 4)² = (7(-13/9) + 17)²

First, let's simplify the expressions inside the parentheses:

  • 2(-13/9) = -26/9
  • -26/9 - 4 = -26/9 - 36/9 = -62/9
  • 7(-13/9) = -91/9
  • -91/9 + 17 = -91/9 + 153/9 = 62/9

Now, substitute these back into the equation:

  • (-62/9)² = (62/9)²
  • 3844/81 = 3844/81

This is true! So, x = -13/9 is indeed a valid solution.

Step 2: Verify x = -21/5

Now let's substitute x = -21/5 into the original equation:

  • (2(-21/5) - 4)² = (7(-21/5) + 17)²

Again, let's simplify the expressions inside the parentheses:

  • 2(-21/5) = -42/5
  • -42/5 - 4 = -42/5 - 20/5 = -62/5
  • 7(-21/5) = -147/5
  • -147/5 + 17 = -147/5 + 85/5 = -62/5

Now, substitute these back into the equation:

  • (-62/5)² = (-62/5)²
  • 3844/25 = 3844/25

This is also true! So, x = -21/5 is a valid solution as well.

Conclusion on Verification

We've successfully verified both potential solutions by plugging them back into the original equation. Both x = -13/9 and x = -21/5 satisfy the equation, meaning they are indeed the correct solutions. Awesome job, guys! We’ve solved the problem completely.

Final Solutions and Summary

Alright, let's wrap things up! We tackled the equation (2x - 4)² = (7x + 17)² using two different methods: expanding and simplifying, and applying the square root property. Both methods led us to the same solutions, which is always a comforting sign. We then took the crucial step of verifying our solutions to ensure their correctness.

The Solutions

The solutions to the equation (2x - 4)² = (7x + 17)² are:

  • x = -13/9
  • x = -21/5

Summary of the Methods

  1. Expanding and Simplifying: This method involves expanding both sides of the equation using the distributive property (or FOIL), rearranging the equation into the standard quadratic form (ax² + bx + c = 0), and then solving for x using factoring or the quadratic formula. This method is a solid, reliable approach that works for many types of quadratic equations.
  2. Using the Square Root Property: This method leverages the property that if a² = b², then a = b or a = -b. It allows us to bypass the expansion step and directly create two linear equations, which are generally easier to solve. This method is particularly efficient when dealing with equations where both sides are perfect squares.

Key Takeaways

  • Understanding the Problem: Before diving into calculations, make sure you understand the structure of the equation and the properties you can use.
  • Multiple Methods: Often, there are multiple ways to solve a math problem. Exploring different methods can deepen your understanding and provide a backup if one method proves difficult.
  • Verification is Crucial: Always, always, always verify your solutions! This simple step can save you from making mistakes and ensures you have the correct answer.
  • Careful Algebra: Be meticulous with your algebraic manipulations, especially when dealing with negative signs and fractions. A small error can lead to an incorrect solution.

So, there you have it! We've successfully solved the equation (2x - 4)² = (7x + 17)², explored different solution methods, and emphasized the importance of verification. Keep practicing, and you'll become a master equation solver in no time! Well done, guys! I hope this was helpful. Keep up the great work! Now go conquer more math problems!