Solving Equations with Floor Functions: A Comprehensive Guide
Solving equations involving floor functions can be quite a puzzle, as it requires a careful balance of algebraic manipulation and careful examination of integer properties. This guide delves into the detailed process of solving the equation lfloor{x^2}rfloor - lfloor{x}rfloor - 42 0 using a step-by-step approach, providing a clear understanding of the concepts and techniques involved.
Introduction to Floor Functions
Before diving into the problem, it's essential to have a clear understanding of floor functions. The floor function, denoted as lfloor x rfloor, rounds a real number down to the nearest integer. This function plays a crucial role in various mathematical problems, including the one we are about to solve.
Problem Statement and Initial Steps
The given equation is:
lfloor{x^2}rfloor - lfloor{x}rfloor - 42 0
To solve this, we first rearrange it to:
lfloor{x^2}rfloor lfloor{x}rfloor 42
Let's denote n lfloor x rfloor. Therefore, we can express x as:
n leq x
Step-by-Step Solution
Step 1: Calculate lfloor x^2 rfloor
Since n leq x , squaring the bounds, we get:
n^2 leq x^2
This implies:
n^2 leq lfloor x^2 rfloor
From our rearranged equation:
lfloor x^2 rfloor n 42
So we need:
n^2 leq n 42
Step 2: Substitute into the Equation
From our rearranged equation, we have:
lfloor x^2 rfloor n 42
Thus, we need:
n^2 leq n 42
Step 3: Solve the Inequalities
First Inequality: n^2 leq n 42
Transform the inequality:
n^2 - n - 42 leq 0
Using the quadratic formula:
n frac{-b pm sqrt{b^2 - 4ac}}{2a}
Substituting a 1, b -1, c -42:
n frac{1 pm sqrt{1 168}}{2} frac{1 pm 13}{2}
This gives us:
n 7 quad text{and} quad n -6
The quadratic n^2 - n - 42 opens upwards, so it is less than or equal to zero between its roots:
-6 leq n leq 7
Second Inequality: n 42
Transform the inequality:
n 42
Thus:
0
Solving:
n^2 n - 41 0
Using the quadratic formula again:
n frac{-1 pm sqrt{1 164}}{2} frac{-1 pm sqrt{165}}{2}
Approximating the roots:
n approx 5.922 quad text{and} quad n approx -6.922
So the relevant intervals are:
n 5
Step 4: Combine the Results
From the first inequality: -6 leq n leq 7
From the second inequality: n or n > 5
The only valid integers that satisfy both conditions are:
n 6, 7
Step 5: Find Corresponding x Values
For n 6:
lfloor x rfloor 6 implies 6 leq x
Then:
lfloor x^2 rfloor 6 42 48
We need:
lfloor x^2 rfloor 48 implies 48 leq x^2
Thus:
sqrt{48} leq x
For n 7:
lfloor x rfloor 7 implies 7 leq x
Then:
lfloor x^2 rfloor 7 42 49
We need:
lfloor x^2 rfloor 49 implies 49 leq x^2
Thus:
7 leq x
Final Answer:
The solution set for x is:
x in [6.928, 7.071)