To solve the problem of finding the integer whose square is closest to the value of [tex]\(\frac{9}{4}\)[/tex], we will proceed step-by-step.
1. Calculate the value of [tex]\(\frac{9}{4}\)[/tex]:
[tex]\[
\frac{9}{4} = 2.25
\][/tex]
2. Determine the square root of [tex]\(2.25\)[/tex]:
To find the integer whose square is approximately equal to [tex]\(2.25\)[/tex], we first find the square root of [tex]\(2.25\)[/tex]:
[tex]\[
\sqrt{2.25} \approx 1.5
\][/tex]
3. Identify the closest integer to [tex]\(1.5\)[/tex]:
The closest integers to [tex]\(1.5\)[/tex] are [tex]\(1\)[/tex] and [tex]\(2\)[/tex]. To decide which is closer, we square these integers and compare the results to [tex]\(2.25\)[/tex]:
- [tex]\(1^2 = 1\)[/tex]
- [tex]\(2^2 = 4\)[/tex]
Comparing these results, [tex]\(2\)[/tex] squared ([tex]\(4\)[/tex]) is closer to [tex]\(2.25\)[/tex] than [tex]\(1\)[/tex] squared ([tex]\(1\)[/tex]).
Therefore, [tex]\(\frac{9}{4}\)[/tex] is close to the square of the integer [tex]\(2\)[/tex], making the closest integer [tex]\(2\)[/tex].
Final answer:
[tex]\[
\boxed{2}
\][/tex]
