Solving For A Number Where The Sum Of It And Its Inverse Equals 3 29/52

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In the realm of mathematics, exploring the relationships between numbers and their inverses often leads to fascinating discoveries and problem-solving opportunities. This discussion delves into a specific scenario the sum of a number and its inverse equals 3 29/52. Understanding this problem requires a blend of algebraic manipulation, quadratic equation solving, and a keen eye for simplification. We will embark on a journey to unravel the intricacies of this mathematical puzzle, providing a comprehensive explanation and a step-by-step solution.

To begin, let's represent the unknown number as 'x'. The inverse of this number would then be 1/x. The problem states that the sum of the number and its inverse is equal to 3 29/52. We can translate this into a mathematical equation as follows:

x + 1/x = 3 29/52

Before we proceed, it's crucial to convert the mixed fraction 3 29/52 into an improper fraction. To do this, we multiply the whole number (3) by the denominator (52) and add the numerator (29), then place the result over the original denominator:

(3 * 52) + 29 = 156 + 29 = 185

So, 3 29/52 is equivalent to 185/52. Our equation now becomes:

x + 1/x = 185/52

This equation is a rational equation, and to solve it, we need to eliminate the fraction. We can do this by multiplying both sides of the equation by 'x':

x * (x + 1/x) = x * (185/52)

Distributing 'x' on the left side, we get:

x^2 + 1 = (185x)/52

To further simplify, we multiply both sides of the equation by 52 to eliminate the denominator:

52 * (x^2 + 1) = 52 * ((185x)/52)

This simplifies to:

52x^2 + 52 = 185x

Now, we need to rearrange the equation into the standard form of a quadratic equation, which is ax^2 + bx + c = 0. To do this, we subtract 185x from both sides of the equation:

52x^2 - 185x + 52 = 0

We now have a quadratic equation in the standard form, where a = 52, b = -185, and c = 52. This equation is the key to finding the value(s) of 'x' that satisfy the original condition.

With the quadratic equation in hand, 52x^2 - 185x + 52 = 0, we can now proceed to solve for 'x'. There are several methods to solve a quadratic equation, including factoring, completing the square, and using the quadratic formula. In this case, factoring might be challenging due to the relatively large coefficients. Therefore, we will employ the quadratic formula, which is a general solution for any quadratic equation in the form ax^2 + bx + c = 0:

x = (-b ± √(b^2 - 4ac)) / (2a)

Here, 'a' is the coefficient of x^2, 'b' is the coefficient of x, and 'c' is the constant term. Substituting the values from our equation, a = 52, b = -185, and c = 52, into the quadratic formula, we get:

x = (-(-185) ± √((-185)^2 - 4 * 52 * 52)) / (2 * 52)

Let's break this down step by step. First, we calculate the discriminant, which is the part under the square root: b^2 - 4ac:

Discriminant = (-185)^2 - 4 * 52 * 52

Calculating this, we have:

(-185)^2 = 34225

4 * 52 * 52 = 4 * 2704 = 10816

So, the discriminant is:

34225 - 10816 = 23409

Now, we take the square root of the discriminant:

√23409 = 153

Substituting these values back into the quadratic formula, we have:

x = (185 ± 153) / (2 * 52)

x = (185 ± 153) / 104

This gives us two possible solutions for 'x':

Now that we have the quadratic formula applied, we find ourselves with two potential solutions for our unknown number, x. This arises from the ± symbol in the formula, indicating that we must consider both the addition and subtraction cases separately. Let's delve into each scenario to determine our two solutions.

Solution 1: Using the Addition Case

In the first case, we'll consider the addition of the square root of the discriminant to 185. This can be expressed as:

x_1 = (185 + 153) / 104

Adding the numbers in the numerator gives us:

185 + 153 = 338

So, our equation becomes:

x_1 = 338 / 104

This fraction can be simplified by finding the greatest common divisor (GCD) of 338 and 104. Both numbers are divisible by 2, giving us:

x_1 = 169 / 52

We can further simplify this fraction by dividing both the numerator and the denominator by 13:

x_1 = 13 / 4

Thus, our first solution is x_1 = 13/4.

Solution 2: Using the Subtraction Case

Now, let's consider the subtraction case, where we subtract the square root of the discriminant from 185. This gives us:

x_2 = (185 - 153) / 104

Subtracting the numbers in the numerator yields:

185 - 153 = 32

So, our equation becomes:

x_2 = 32 / 104

Again, we can simplify this fraction by finding the GCD of 32 and 104. Both numbers are divisible by 8, so we divide both the numerator and the denominator by 8:

x_2 = 4 / 13

Therefore, our second solution is x_2 = 4/13.

Having obtained two potential solutions, x_1 = 13/4 and x_2 = 4/13, it is imperative that we verify whether these values indeed satisfy the original equation. This step is crucial as it confirms the accuracy of our calculations and ensures that the solutions are valid within the context of the problem. To verify, we will substitute each solution back into the original equation, x + 1/x = 185/52, and check if the equation holds true.

Verification of Solution 1: x_1 = 13/4

Let's substitute x_1 = 13/4 into the original equation:

(13/4) + 1/(13/4) = 185/52

To simplify the second term, we take the reciprocal of 13/4, which is 4/13. Our equation now becomes:

(13/4) + (4/13) = 185/52

To add the fractions on the left side, we need a common denominator, which is the least common multiple (LCM) of 4 and 13. The LCM of 4 and 13 is 52. We rewrite the fractions with the common denominator:

(13/4) * (13/13) = 169/52

(4/13) * (4/4) = 16/52

Now we can add the fractions:

(169/52) + (16/52) = 185/52

This simplifies to:

185/52 = 185/52

As the equation holds true, we can confirm that x_1 = 13/4 is a valid solution.

Verification of Solution 2: x_2 = 4/13

Now, let's substitute x_2 = 4/13 into the original equation:

(4/13) + 1/(4/13) = 185/52

Again, we simplify the second term by taking the reciprocal of 4/13, which is 13/4. Our equation now becomes:

(4/13) + (13/4) = 185/52

We already know from our verification of x_1 that the sum of these fractions is 185/52. So, we can directly write:

185/52 = 185/52

Since this equation also holds true, we confirm that x_2 = 4/13 is also a valid solution.

In conclusion, we embarked on a mathematical journey to solve a problem involving the sum of a number and its inverse. We transformed the given condition into a quadratic equation, applied the quadratic formula, and successfully found two solutions: 13/4 and 4/13. Moreover, we meticulously verified these solutions by substituting them back into the original equation, affirming their validity. This exercise not only demonstrates the power of algebraic manipulation and problem-solving techniques but also highlights the inherent beauty and interconnectedness within the realm of mathematics. The ability to approach complex problems methodically, break them down into manageable steps, and apply appropriate techniques is a testament to the enduring importance of mathematical reasoning. As we conclude this discussion, it is our hope that the step-by-step approach outlined here provides a solid foundation for tackling similar problems and inspires further exploration into the fascinating world of mathematics.