Solving For X - 17/√x Given X - √x = 17

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Hey guys! Let's dive into this interesting math problem where we need to find the value of the expression x - 17/√x, given that x - √x = 17. This problem looks tricky at first, but we'll break it down step by step to make it super clear. We'll explore different algebraic manipulations and techniques to arrive at the solution. So, buckle up and let's get started!

Understanding the Problem

Before we jump into solving, let's make sure we fully understand what the problem is asking. We are given the equation x - √x = 17, and our goal is to find the value of the expression x - 17/√x. The key here is to somehow relate the given equation to the expression we want to find. This often involves algebraic manipulation and clever substitutions. Think of it like a puzzle where we need to rearrange the pieces to fit together. We need to find a way to use the information we have to get to the answer we want. So, let's start exploring some ways to manipulate the given equation and see where it leads us. Remember, in math, sometimes the journey of finding the solution is just as important as the solution itself!

Initial Algebraic Manipulations

The first step in tackling this problem is to try and manipulate the given equation, x - √x = 17, into a more usable form. A common technique when dealing with square roots is to try and isolate the square root term. So, let's add √x to both sides of the equation. This gives us x = √x + 17. Now, we have √x isolated on one side. This is a good start because it allows us to see the relationship between x and √x more clearly.

Next, we can try to get rid of the square root by squaring both sides of the equation. Squaring both sides of x = √x + 17 gives us x² = (√x + 17)². Expanding the right side, we get x² = x + 34√x + 289. This might look more complicated, but it's a crucial step. We've eliminated one square root, but introduced a new one in the process. Don't worry; this is part of the process. Our goal is to find a relationship that helps us evaluate x - 17/√x, and this manipulation is bringing us closer. Remember, algebraic manipulations are like tools in our toolbox – we use them to reshape the problem into something we can handle more easily. The key is to keep experimenting and trying different approaches until we find the one that works!

Substitution to Simplify the Equation

To further simplify the equation, let's use a substitution. This is a powerful technique in algebra that can make complex expressions more manageable. Let's substitute y = √x. This means that y² = x. Now, we can rewrite our original equation x - √x = 17 in terms of y. Replacing x with y² and √x with y, we get y² - y = 17.

This quadratic equation looks much simpler to work with! We can rearrange it to the standard form of a quadratic equation: y² - y - 17 = 0. Now, we have a familiar form that we can solve using the quadratic formula or by completing the square. This substitution has transformed our original equation into a more manageable form, which is a common strategy in problem-solving. By making the right substitution, we can often simplify the problem and make it easier to find a solution. So, let's keep this simplified equation in mind as we move forward and explore how to solve for y.

Solving the Quadratic Equation

Now that we have the quadratic equation y² - y - 17 = 0, let's solve for y. We can use the quadratic formula, which is a reliable method for finding the roots of any quadratic equation in the form ay² + by + c = 0. The quadratic formula is given by: y = (-b ± √(b² - 4ac)) / (2a). In our case, a = 1, b = -1, and c = -17. Plugging these values into the quadratic formula, we get:

y = (1 ± √((-1)² - 4 * 1 * (-17))) / (2 * 1)

Simplifying further:

y = (1 ± √(1 + 68)) / 2

y = (1 ± √69) / 2

So, we have two possible values for y: y = (1 + √69) / 2 and y = (1 - √69) / 2. However, since y = √x, y must be non-negative. The second solution, (1 - √69) / 2, is negative because √69 is greater than 1. Therefore, we can discard the negative solution. This leaves us with y = (1 + √69) / 2. This is a crucial step because we've now found the value of y, which is equal to √x. With this value in hand, we can move closer to finding the value of the expression x - 17/√x.

Finding x and √x

From the previous step, we found that y = √x = (1 + √69) / 2. Now, to find x, we simply need to square y: x = y² = ((1 + √69) / 2)². Expanding this, we get:

x = (1 + 2√69 + 69) / 4

x = (70 + 2√69) / 4

x = (35 + √69) / 2

Now we have the value of x. We also know the value of √x, which is (1 + √69) / 2. These values are crucial for finding the value of the expression x - 17/√x. We've essentially broken down the problem into smaller, more manageable parts. We found √x by solving the quadratic equation, and then we found x by squaring √x. Now, all that's left is to plug these values into the expression we want to evaluate.

Evaluating the Expression x - 17/√x

Now that we have the values of x and √x, we can finally evaluate the expression x - 17/√x. We found that x = (35 + √69) / 2 and √x = (1 + √69) / 2. Plugging these values into the expression, we get:

x - 17/√x = ((35 + √69) / 2) - 17 / ((1 + √69) / 2)

To simplify this, we can multiply the numerator and denominator of the second term by 2:

= ((35 + √69) / 2) - (34 / (1 + √69))

Next, we rationalize the denominator of the second term by multiplying both the numerator and denominator by the conjugate of (1 + √69), which is (1 - √69):

= ((35 + √69) / 2) - (34(1 - √69) / ((1 + √69)(1 - √69)))

= ((35 + √69) / 2) - (34(1 - √69) / (1 - 69))

= ((35 + √69) / 2) - (34(1 - √69) / (-68))

= ((35 + √69) / 2) + ((1 - √69) / 2)

Now, we can combine the terms:

= (35 + √69 + 1 - √69) / 2

= 36 / 2

= 18

So, the value of the expression x - 17/√x is 18. We've successfully navigated through the algebraic manipulations, substitutions, and simplifications to arrive at our final answer! This problem demonstrates the power of breaking down a complex problem into smaller, more manageable steps. By using techniques like substitution and rationalizing the denominator, we were able to solve for the value of the expression.

Conclusion

In conclusion, given x - √x = 17, the value of the expression x - 17/√x is 18. We arrived at this solution by first manipulating the given equation to isolate the square root term. Then, we used a substitution to simplify the equation into a quadratic form. We solved the quadratic equation using the quadratic formula, and from there, we found the values of x and √x. Finally, we plugged these values into the expression x - 17/√x and simplified to get our answer.

This problem highlights the importance of algebraic manipulation, substitution, and simplification in solving complex mathematical problems. By breaking down the problem into smaller steps and applying appropriate techniques, we can make even the most challenging problems manageable. Remember, practice makes perfect, so keep exploring different problem-solving strategies and techniques. You'll be surprised at how much you can achieve!