If x is the smallest number which is to be subtracted from 2605 to make it a perfect square, then the
value of (4x +7) is :

Q: If x is the smallest number which is to be subtracted from 2605 to make it a perfect square, then the value of (4x +7) is :

To determine the smallest number $ x $ that needs to be subtracted from 2605 to make it a perfect square, we first need to find the nearest perfect square less than 2605. 1. **Finding the nearest perfect square:** - Calculate the square root of 2605: $\sqrt{2605} \approx 51.04$. - The nearest whole number less than 51.04 is 51. - Calculate the square of 51: $51^2 = 2601$. 2. **Determine $ x $:** - Subtract 2601 from 2605 to find $ x $: $2605 - 2601 = 4$. - Therefore, $ x = 4 $. 3. **Calculate $ 4x + 7 $:** - Substitute $ x = 4 $ into the expression $ 4x + 7 $: - $ 4 \times 4 + 7 = 16 + 7 = 23 $. Thus, the value of $ 4x + 7 $ is 23, which corresponds to Option 3. **Explanation for why other options are incorrect:** - **Option 1 (31):** - If $ 4x + 7 = 31 $, then $ 4x = 31 - 7 = 24 $, leading to $ x = 6 $. - Subtracting 6 from 2605 gives 2599, which is not a perfect square. - **Option 2 (11):** - If $ 4x + 7 = 11 $, then $ 4x = 11 - 7 = 4 $, leading to $ x = 1 $. - Subtracting 1 from 2605 gives 2604, which is not a perfect square. - **Option 4 (27):** - If $ 4x + 7 = 27 $, then $ 4x = 27 - 7 = 20 $, leading to $ x = 5 $. - Subtracting 5 from 2605 gives 2600, which is not a perfect square. Therefore, Option 3 is the only correct choice, as it results in a perfect square when $ x $ is subtracted from 2605.

Question

Accepted Answer

To determine the smallest number $ x $ that needs to be subtracted from 2605 to make it a perfect square, we first need to find the nearest perfect square less than 2605.

1. **Finding the nearest perfect square:**
   - Calculate the square root of 2605: $\sqrt{2605} \approx 51.04$.
   - The nearest whole number less than 51.04 is 51.
   - Calculate the square of 51: $51^2 = 2601$.

2. **Determine $ x $:**
   - Subtract 2601 from 2605 to find $ x $: $2605 - 2601 = 4$.
   - Therefore, $ x = 4 $.

3. **Calculate $ 4x + 7 $:**
   - Substitute $ x = 4 $ into the expression $ 4x + 7 $:
   - $ 4 \times 4 + 7 = 16 + 7 = 23 $.

Thus, the value of $ 4x + 7 $ is 23, which corresponds to Option 3.

**Explanation for why other options are incorrect:**

- **Option 1 (31):** 
  - If $ 4x + 7 = 31 $, then $ 4x = 31 - 7 = 24 $, leading to $ x = 6 $.
  - Subtracting 6 from 2605 gives 2599, which is not a perfect square.

- **Option 2 (11):**
  - If $ 4x + 7 = 11 $, then $ 4x = 11 - 7 = 4 $, leading to $ x = 1 $.
  - Subtracting 1 from 2605 gives 2604, which is not a perfect square.

- **Option 4 (27):**
  - If $ 4x + 7 = 27 $, then $ 4x = 27 - 7 = 20 $, leading to $ x = 5 $.
  - Subtracting 5 from 2605 gives 2600, which is not a perfect square.

Therefore, Option 3 is the only correct choice, as it results in a perfect square when $ x $ is subtracted from 2605.