Solving for x/1 in a Complex Expression

When dealing with complex expressions involving square roots, rationalization and algebraic manipulation can provide the necessary tools to simplify these expressions. This article explores a specific example, demonstrating the step-by-step process of finding the value of x/1 when x 7 - 4√3. The focus is on understanding the techniques of rationalization and quadratic equations to solve such problems effectively.

The Problem at Hand

To find the value of x/1 when x 7 - 4√3, we need to first determine 1/x. The steps involved in this calculation will be detailed and explained in a straightforward manner, ensuring a clear understanding of the process.

Step 1: Rationalizing the Denominator

The first step is to rationalize the denominator of the expression 1/(7 - 4√3). This involves multiplying both the numerator and the denominator by the conjugate of the denominator, 7 4√3.

1/(7 - 4√3) (7 4√3) / ((7 - 4√3)(7 4√3))

The denominator becomes a difference of squares, simplifying to:

(7 - 4√3)(7 4√3) 72 - (4√3)2 49 - 48 1

Therefore,

1/(7 - 4√3) 7 4√3

Step 2: Finding x/1

Now, we substitute x 7 - 4√3 and 1/x 7 4√3 into the expression x/1.

x/1 (7 - 4√3)(7 4√3)

Using the same difference of squares principle, we get:

(7 - 4√3)(7 4√3) 72 - (4√3)2 49 - 48 1

Thus,

x/1 14

The Quadratic Equation Insight

Another way to approach this problem is by recognizing that x and its conjugate, x? 7 4√3, satisfy the quadratic equation x2 - 14x 1 0. From this, we can see that the sum of the roots (14) gives us the required value without further calculation.

Let x and x? be the roots. From the quadratic equation, we know:

x x? 14 and x * x? 1

This means x/1 14

Further Explorations

To solidify the understanding, let's consider a couple of examples for similar problems:

Example 1: x 8 - 4√3

In this case, the conjugate is x? 8 4√3. Using the same principle:

1/(8 - 4√3) (8 4√3) / ((8 - 4√3)(8 4√3)) and simplifying, we get:

(8 - 4√3)(8 4√3) 64 - 48 16

x/1 (8 - 4√3)(8 4√3) 64 - 48 16

Therefore, x/1 16

Example 2: x 9 - 4√5

Here, the conjugate is x? 9 4√5. Following the same procedure:

1/(9 - 4√5) (9 4√5) / ((9 - 4√5)(9 4√5)) and simplifying, we obtain:

(9 - 4√5)(9 4√5) 81 - 80 1

x/1 (9 - 4√5)(9 4√5) 81 - 80 1

Therefore, x/1 18

These examples illustrate the general approach to solving such problems, emphasizing the importance of rationalization and the properties of quadratic equations.