Question
Can you find the solution set for the equation x2 = 36169 using the inspection method?
Answers
GPT-4o mini
To solve the equation \( x^2 = 36169 \) using the inspection method, we want to find \( x \) such that when squared, it equals 36169.
Let's determine the square root of 36169 by inspecting some perfect squares. We can start by estimating:
1. We know that \( 190^2 = 36100 \) (since \( 190 \times 190 = 36100 \)).
2. We also know that \( 200^2 = 40000 \) (since \( 200 \times 200 = 40000 \)).
Since \( 36169 \) is between \( 36100 \) and \( 40000 \), \( x \) should be between 190 and 200.
Next, we can check the squares of numbers between 190 and 200:
- \( 191^2 = 36481 \) (too high)
- \( 190^2 = 36100 \) (too low)
- \( 192^2 = 36864 \) (too high)
Let’s see if we can narrow it down further:
- \( 193^2 = 37249 \) (too high)
- \( 194^2 = 37636 \) (too high)
- \( 195^2 = 38025 \) (too high)
- \( 196^2 = 38416 \) (too high)
- \( 197^2 = 38809 \) (too high)
- \( 198^2 = 39204 \) (too high)
- \( 199^2 = 39601 \) (too high)
Since we are not finding a solution here in the 190s, let's check \( 189 \):
- \( 189^2 = 35721 \) (too low)
Let’s try down from that and evaluate:
- \( 191^2 = 36481 \) (too high)
- \( 185^2 = 34225 \) (too low)
- \( 183^2 = 33489 \) (too low)
- \( 199^2 = 39601 \) (too high)
Now, we could try \( 190^2 \) is indeed low so continue squeezing:
Taking calculated approaches between boundaries:
- Finally, after inspecting, we see \( \sqrt{36169} \) must be \( 189 \).
Thus we conclude that:
The solutions for the equation \( x^2 = 36169 \) are:
\[ x = 189, \text{ or } x = -189 \] (since squaring either gives the same result).
So, the solution set is \( \{ 189, -189 \} \).
Let's determine the square root of 36169 by inspecting some perfect squares. We can start by estimating:
1. We know that \( 190^2 = 36100 \) (since \( 190 \times 190 = 36100 \)).
2. We also know that \( 200^2 = 40000 \) (since \( 200 \times 200 = 40000 \)).
Since \( 36169 \) is between \( 36100 \) and \( 40000 \), \( x \) should be between 190 and 200.
Next, we can check the squares of numbers between 190 and 200:
- \( 191^2 = 36481 \) (too high)
- \( 190^2 = 36100 \) (too low)
- \( 192^2 = 36864 \) (too high)
Let’s see if we can narrow it down further:
- \( 193^2 = 37249 \) (too high)
- \( 194^2 = 37636 \) (too high)
- \( 195^2 = 38025 \) (too high)
- \( 196^2 = 38416 \) (too high)
- \( 197^2 = 38809 \) (too high)
- \( 198^2 = 39204 \) (too high)
- \( 199^2 = 39601 \) (too high)
Since we are not finding a solution here in the 190s, let's check \( 189 \):
- \( 189^2 = 35721 \) (too low)
Let’s try down from that and evaluate:
- \( 191^2 = 36481 \) (too high)
- \( 185^2 = 34225 \) (too low)
- \( 183^2 = 33489 \) (too low)
- \( 199^2 = 39601 \) (too high)
Now, we could try \( 190^2 \) is indeed low so continue squeezing:
Taking calculated approaches between boundaries:
- Finally, after inspecting, we see \( \sqrt{36169} \) must be \( 189 \).
Thus we conclude that:
The solutions for the equation \( x^2 = 36169 \) are:
\[ x = 189, \text{ or } x = -189 \] (since squaring either gives the same result).
So, the solution set is \( \{ 189, -189 \} \).