Problem Statement:

A natural number, when increased by 84, equals 160 times its reciprocal. Find the number.

Solution:

Let's assume the natural number we are trying to find is "x".

Step 1: Formulating the equation

According to the problem statement, the number increased by 84 is equal to 160 times its reciprocal. Mathematically, we can represent this as:

x + 84 = 160 * (1/x)

Step 2: Simplifying the equation

To simplify the equation, let's first remove the fraction by multiplying both sides of the equation by "x". This gives us:

x * (x + 84) = 160

Expanding the equation, we get:

x^2 + 84x = 160

Step 3: Solving the quadratic equation

To solve the quadratic equation, we need to rearrange it into the standard form "ax^2 + bx + c = 0". In this case, the equation becomes:

x^2 + 84x - 160 = 0

Now we can solve this quadratic equation using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In our case, a = 1, b = 84, and c = -160. Plugging these values into the formula, we get:

x = (-84 ± √(84^2 - 4*1*(-160))) / (2*1)

Simplifying further, we have:

x = (-84 ± √(7056 + 640)) / 2

x = (-84 ± √(7696)) / 2

x = (-84 ± 88) / 2

Step 4: Finding the solutions

Using the quadratic formula, we have two possible solutions for x:

1. x = (-84 + 88) / 2 = 4 / 2 = 2
2. x = (-84 - 88) / 2 = -172 / 2 = -86

However, since we are looking for a natural number, the second solution (-86) is not valid. Therefore, the natural number we are looking for is 2.

Conclusion:

The natural number that, when increased by 84, equals 160 times its reciprocal, is 2.

Use discrimant formula..-42 - 2 root 481