Solve: The Square Of (x-9) Is X-3. Find X!

Hey there, math enthusiasts! Ever stumbled upon a word problem that felt like unlocking a secret code? Well, let's dive into one today that's got a bit of algebraic flair. We're going to tackle the question: What number, when 9 less than it is squared, results in a value that is 3 less than the original number? Sounds intriguing, right? Let's break it down step by step.

Translating Words into Equations

The first step in solving any word problem is to translate the words into a mathematical equation. This might seem daunting, but it's like learning a new language – once you understand the grammar, you can decipher almost anything! In this problem, our key is to identify the relationships between the numbers and operations described.

Let’s use 'x' to represent the mystery number we're trying to find. The problem states "9 less than a number," which we can write as (x - 9). Then, it says we need to square this, giving us (x - 9)². The phrase "3 less than the number" translates to (x - 3). Finally, the problem tells us that the square of (x - 9) is equal to (x - 3). Putting it all together, we get the equation:

(x - 9)² = x - 3

This equation is the heart of our problem. Now that we have it, we can use our algebra skills to solve for 'x'. Remember, equations are like balanced scales – what we do to one side, we must do to the other to keep things equal.

Unpacking the Equation: Expanding and Simplifying

Now that we have our equation, (x - 9)² = x - 3, the next step is to expand and simplify it. This will help us get the equation into a more manageable form, specifically a quadratic equation. Don't worry, it's not as scary as it sounds! We'll take it one step at a time.

First, let's tackle the left side of the equation, (x - 9)². Remember that squaring a binomial means multiplying it by itself: (x - 9)² = (x - 9)(x - 9). To multiply this, we can use the FOIL method (First, Outer, Inner, Last) or the distributive property. Let's use the distributive property here:

• x(x - 9) - 9(x - 9)
• = x² - 9x - 9x + 81
• = x² - 18x + 81

So, (x - 9)² expands to x² - 18x + 81. Now, we can substitute this back into our original equation:

x² - 18x + 81 = x - 3

Next, we want to get all the terms on one side of the equation, leaving zero on the other side. This is a crucial step in solving quadratic equations. To do this, we subtract 'x' and add '3' to both sides of the equation:

• x² - 18x + 81 - x + 3 = x - 3 - x + 3
• x² - 19x + 84 = 0

Now we have a simplified quadratic equation in the standard form: x² - 19x + 84 = 0. This form is perfect for factoring, which is our next step.

The Art of Factoring: Finding the Roots

Now that we have our simplified quadratic equation, x² - 19x + 84 = 0, it's time to factor! Factoring is like reverse-engineering multiplication. We want to find two binomials that, when multiplied together, give us our quadratic equation. This might sound tricky, but with a little practice, it becomes second nature.

To factor x² - 19x + 84, we need to find two numbers that:

• Multiply to 84 (the constant term)
• Add up to -19 (the coefficient of the x term)

Let's think about the factors of 84. We have pairs like (1, 84), (2, 42), (3, 28), (4, 21), (6, 14), and (7, 12). Since we need the numbers to add up to -19, we know both numbers must be negative. Looking at our pairs, -7 and -12 fit the bill perfectly: (-7) * (-12) = 84 and (-7) + (-12) = -19.

So, we can factor our quadratic equation as:

(x - 7)(x - 12) = 0

This factored form is incredibly useful because it allows us to easily find the solutions for 'x'. Remember the zero-product property? It states that if the product of two factors is zero, then at least one of the factors must be zero. In our case, this means either (x - 7) = 0 or (x - 12) = 0.

Solving these simple equations gives us our possible solutions:

• x - 7 = 0 => x = 7
• x - 12 = 0 => x = 12

So, we have two potential answers: 7 and 12. But before we declare victory, we need to check if both solutions actually work in our original equation. This is a crucial step to avoid extraneous solutions.

The Moment of Truth: Verifying the Solutions

We've arrived at the exciting stage of our mathematical journey: verifying our solutions! We found two potential answers for 'x': 7 and 12. But not every solution that pops out of our equations is necessarily a true solution to the original problem. Sometimes, we encounter what are called extraneous solutions, which satisfy the transformed equation but not the initial one. So, let's put our detective hats on and check each answer.

Our original equation, if you recall, was (x - 9)² = x - 3. To verify our solutions, we'll substitute each value of 'x' back into this equation and see if both sides are equal.

Let's start with x = 7:

• (7 - 9)² = 7 - 3
• (-2)² = 4
• 4 = 4

It checks out! When x = 7, both sides of the equation are equal. This means 7 is indeed a valid solution to our problem.

Now, let's test x = 12:

• (12 - 9)² = 12 - 3
• (3)² = 9
• 9 = 9

Fantastic! When x = 12, the equation also holds true. This confirms that 12 is another valid solution.

So, after our thorough investigation, we can confidently say that both 7 and 12 are solutions to the equation (x - 9)² = x - 3. But what does this mean in the context of our original word problem? Let's revisit the question.

The Grand Finale: Answering the Question

We've reached the final stage of our mathematical quest, and it's time to answer the original question: What number, when 9 less than it is squared, results in a value that is 3 less than the original number? We've journeyed through translating the words into an equation, expanding and simplifying, factoring, and verifying solutions. Now, we get to reap the rewards of our hard work.

Through our algebraic explorations, we discovered that there are not one, but two numbers that satisfy the conditions of the problem: 7 and 12. Let's take a moment to appreciate this. Often in mathematics, problems have a single, neat solution. But sometimes, like in this case, there are multiple answers that fit the criteria. This is part of what makes math so fascinating – it's full of surprises!

To ensure we've truly grasped the solution, let's revisit the original problem and see how our answers fit.

• For the number 7: 9 less than 7 is -2. Squaring -2 gives us 4. And indeed, 4 is 3 less than 7.
• For the number 12: 9 less than 12 is 3. Squaring 3 gives us 9. And yes, 9 is 3 less than 12.

Both solutions perfectly align with the problem's conditions. We've successfully navigated the twists and turns of this mathematical puzzle and arrived at a satisfying conclusion.

Conclusion

So, there you have it! We've unraveled the mystery of the number (or numbers!) that fit the given conditions. Remember, the key to solving word problems lies in breaking them down step by step, translating the words into mathematical language, and applying the right algebraic tools. And don't forget to verify your solutions – it's like the final flourish on a masterpiece!

If you're eager to explore more mathematical puzzles and challenges, there are fantastic resources available online. A great place to start is Khan Academy's Algebra Section, where you can find a wealth of tutorials, practice problems, and explanations to further enhance your algebraic prowess. Keep practicing, keep exploring, and most importantly, keep enjoying the fascinating world of mathematics!