Solving The Equation X - 9 = √(x - 7) A Step-by-Step Guide

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Hey guys! Ever stumbled upon an equation that looks like it's straight out of a math puzzle? Today, we're going to tackle one of those head-scratchers together: solving the equation x - 9 = √( x - 7). Don't worry, it might seem intimidating at first, but we'll break it down into simple, manageable steps. So, grab your thinking caps, and let's dive into the world of algebra!

Understanding the Equation

Before we jump into the solution, let's make sure we understand what we're dealing with. The equation x - 9 = √( x - 7) involves a variable (x), subtraction, and a square root. Our goal is to find the value(s) of x that make this equation true. This type of equation, where the variable is under a square root, is called a radical equation. Solving radical equations requires a few special techniques, so let's get started!

First off, you need to understand that you're looking for a value, or potentially values, of 'x' that will make the equation balance perfectly. Think of it like a scale – both sides need to weigh the same. On one side, we have 'x' being reduced by 9, and on the other side, we have the square root of 'x' being reduced by 7. Our mission, should we choose to accept it, is to find the number that makes this balance work.

But why does this matter, you might ask? Well, solving equations like this isn't just some abstract math exercise. It’s actually super useful in all sorts of real-world situations. Imagine you're designing a bridge, calculating the trajectory of a rocket, or even figuring out the best way to invest your money. Math, and algebra in particular, forms the backbone of so many calculations and predictions. So, mastering these skills now is going to pay off big time later!

Step 1: Isolate the Square Root

The first crucial step in solving the equation is to isolate the square root term. In our case, √( x - 7) is already isolated on the right side of the equation. This is excellent news because it means we can move directly to the next step. However, in other problems, you might need to perform some algebraic manipulations (like adding or subtracting terms) to get the square root term by itself. Remember, the goal here is to get the radical term alone on one side of the equation. This sets us up perfectly for the next move, which involves getting rid of that pesky square root symbol.

Think of isolating the square root like setting the stage for the main event. You wouldn't want distractions cluttering the view, right? Similarly, we want the square root term to be the star of the show, with no other operations messing things up on its side of the equation. Sometimes, this might involve adding a number to both sides, subtracting a term, or even multiplying or dividing. The key is to use inverse operations – doing the opposite of what's currently being done – to gradually peel away the layers until the square root stands alone and proud. This isolation process is a fundamental technique in algebra, and mastering it will make your equation-solving life much, much easier. So, keep practicing, and you'll become a pro in no time!

Step 2: Square Both Sides

Now comes the fun part: getting rid of the square root! To do this, we square both sides of the equation. Squaring both sides is a valid operation because it maintains the equality, as long as we do it to both sides. When we square the left side, (x - 9)², we get (x - 9)(x - 9), which we'll expand later. When we square the right side, (√( x - 7))², the square root and the square cancel each other out, leaving us with just x - 7. This is the magic of solving the equation! We've transformed a radical equation into a more manageable quadratic equation.

Imagine the square root as a shield, protecting the expression underneath. Squaring both sides is like using a special weapon that breaks that shield, freeing the expression inside. But, and this is a big but, you have to make sure you hit both sides of the equation equally! Think of it like balancing a seesaw – if you add weight to one side, you need to add the same weight to the other to keep it level. Squaring both sides ensures that our equation remains balanced, and that we're still on the right track to finding the correct solution.

This step is so important because it allows us to move from the realm of radical equations, which can be tricky to deal with, to the more familiar territory of polynomials. Polynomials, especially quadratics, have a whole arsenal of techniques we can use to solve them, from factoring to the quadratic formula. So, by squaring both sides, we've essentially opened up a whole new world of possibilities for finding our value of 'x'. Just remember to be careful with your algebra as you expand and simplify – those pesky signs and exponents can sometimes trip you up if you're not paying attention!

Step 3: Expand and Simplify

After squaring both sides, we have (x - 9)² = x - 7. Now, we need to expand the left side. Remember, (x - 9)² means (x - 9)(x - 9). Using the FOIL method (First, Outer, Inner, Last) or the distributive property, we get:

  • x² - 9x - 9x + 81 = x - 7

Combining like terms, we have:

  • x² - 18x + 81 = x - 7

Next, we want to set the equation to zero, which is a standard step when solving the equation quadratic equations. To do this, we subtract x and add 7 to both sides:

  • x² - 18x - x + 81 + 7 = 0

  • x² - 19x + 88 = 0

Great! We now have a quadratic equation in the standard form ax² + bx + c = 0.

Think of expanding and simplifying as cleaning up your workspace before you start a big project. You wouldn't want to try building a skyscraper on a messy construction site, would you? Similarly, we need to tidy up our equation so that it's in a form we can easily work with. Expanding removes the parentheses and exponents, while simplifying combines like terms and puts the equation in its most basic form.

The FOIL method, which stands for First, Outer, Inner, Last, is a handy tool for expanding binomials (expressions with two terms) like (x - 9)(x - 9). It's a systematic way of ensuring you multiply each term in the first binomial by each term in the second binomial. But remember, the distributive property is the real hero here – it's the underlying principle that makes FOIL work. If you ever forget the FOIL shortcut, just remember to distribute each term, and you'll be golden!

Setting the equation to zero is another crucial step in solving the equation quadratic equations. This is because many of the techniques we use, like factoring and the quadratic formula, rely on having the equation in this form. It's like setting a common language for all our solving tools – they all understand how to work with an equation that equals zero. So, keep this in mind as you tackle quadratic equations – setting to zero is your first step towards unlocking the solution!

Step 4: Solve the Quadratic Equation

Now we need to solve the quadratic equation x² - 19x + 88 = 0. There are a couple of ways we can do this: factoring or using the quadratic formula. Let's try factoring first. We're looking for two numbers that multiply to 88 and add up to -19. After a little thought, we can see that -8 and -11 fit the bill since (-8) * (-11) = 88 and (-8) + (-11) = -19. So, we can factor the quadratic equation as:

  • (x - 8)(x - 11) = 0

Now, we use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This means either x - 8 = 0 or x - 11 = 0. Solving the equation each of these linear equations gives us:

  • x = 8 or x = 11

So, we have two potential solutions: x = 8 and x = 11.

Solving a quadratic equation can sometimes feel like cracking a code, but once you understand the tools and techniques, it becomes a lot less mysterious. Factoring is like finding the hidden keys that unlock the equation's secrets. It involves breaking down the quadratic expression into two binomial factors, which then allows us to use the zero-product property to find the solutions.

But what if factoring doesn't work? That's where the quadratic formula comes in! This trusty formula is like a universal key that can unlock any quadratic equation, no matter how complex. It might look a little intimidating at first, with its square roots and fractions, but once you learn how to plug in the coefficients and simplify, it becomes a powerful weapon in your math arsenal.

Whether you choose factoring or the quadratic formula, the key is to practice, practice, practice! The more you work with quadratic equations, the more comfortable you'll become with the different methods, and the faster you'll be able to find the solutions. So, don't be afraid to tackle those problems head-on, and remember, every solved equation is a victory!

Step 5: Check for Extraneous Solutions

This is a crucial step that's often overlooked, but it's super important when solving the equation radical equations! Because we squared both sides of the equation, we might have introduced extraneous solutions. These are solutions that we found algebraically, but they don't actually satisfy the original equation. To check, we need to plug each potential solution back into the original equation, x - 9 = √( x - 7), and see if it holds true.

Let's start with x = 8:

  • 8 - 9 = √(8 - 7)

  • -1 = √1

  • -1 = 1

This is false, so x = 8 is an extraneous solution.

Now let's check x = 11:

  • 11 - 9 = √(11 - 7)

  • 2 = √4

  • 2 = 2

This is true, so x = 11 is a valid solution.

Therefore, the only solution to the equation x - 9 = √( x - 7) is x = 11.

Checking for extraneous solutions is like double-checking your work before submitting it. You wouldn't want to turn in a paper full of mistakes, right? Similarly, we need to make sure our solutions actually work in the original equation. Squaring both sides, while a powerful technique, can sometimes lead us astray, introducing solutions that don't belong. These are the sneaky extraneous solutions, and they're waiting to trick us if we're not careful!

Think of the original equation as a gatekeeper, only allowing the true solutions to pass through. Extraneous solutions are like imposters, trying to sneak in without the proper credentials. Plugging our potential solutions back into the original equation is like showing them to the gatekeeper and seeing if they have the right ID. If they don't, they get rejected, and we know they're not valid solutions.

This step is especially important in radical equations, where the square root can sometimes mask the true nature of the solution. So, always remember to check for extraneous solutions, and you'll be sure to find the correct answer every time!

Conclusion

And there you have it! We've successfully solved the equation x - 9 = √( x - 7). We isolated the square root, squared both sides, expanded and simplified, solved the quadratic equation, and, most importantly, checked for extraneous solutions. The only valid solution is x = 11. Keep practicing these steps, and you'll become a master at solving the equation radical equations. Keep up the great work, guys! You're doing awesome!