A person needs to arrange desks in a classroom such that the number of rows is the same
as the number of columns of desks. After having made such an arrangement, he found that
34 desks were still left with him. When he doubled the number of rows and columns, he
found that he was short of 14 desks. Find the total number of desks available with him.
Show your work.


Let's solve this step by step by setting up equations based on the given conditions.

### Step 1: Defining the variables
Let:
- \( x \) be the number of rows and columns in the original arrangement. Since the rows and columns are equal, the total number of desks arranged will be \( x^2 \).

### Step 2: First condition
It is given that 34 desks were left after arranging the desks in a square grid. This means the total number of desks is \( x^2 + 34 \). So the equation for the total number of desks based on this condition is:

\[
N = x^2 + 34 \tag{1}
\]

where \( N \) is the total number of desks.

### Step 3: Second condition
It is also given that when the number of rows and columns is doubled (i.e., the new arrangement is \( 2x \times 2x \), making the total number of desks \( 4x^2 \)), the person is short of 14 desks. This means the total number of desks is \( 4x^2 - 14 \). So the equation for the total number of desks based on this condition is:

\[
N = 4x^2 - 14 \tag{2}
\]

### Step 4: Solving the system of equations
Now we have two equations:

1. \( N = x^2 + 34 \)
2. \( N = 4x^2 - 14 \)

Equating both expressions for \( N \):

\[
x^2 + 34 = 4x^2 - 14
\]

### Step 5: Simplifying the equation
Move all terms involving \( x \) to one side and constants to the other:

\[
34 + 14 = 4x^2 - x^2
\]

\[
48 = 3x^2
\]

### Step 6: Solving for \( x \)
Divide both sides by 3:

\[
x^2 = \frac{48}{3} = 16
\]

Take the square root of both sides:

\[
x = 4
\]

### Step 7: Finding the total number of desks
Now that we know \( x = 4 \), substitute this value back into either equation for \( N \). Using equation (1):

\[
N = x^2 + 34 = 4^2 + 34 = 16 + 34 = 50
\]

Thus, the total number of desks available is **50**.

### Final Answer:
The person has **50 desks** in total.

---

Would you like a breakdown of any step? Here are some related questions to explore:

1. What if 40 desks were left instead of 34? How would that change the result?
2. How would the problem change if the number of columns was different from the rows?
3. Can you derive an equation for the case where there are still \( k \) desks left after arranging the desks?
4. How many desks would the person need if the rows and columns were tripled instead of doubled?
5. How does changing the shortage of desks from 14 to 20 affect the result?

**Tip:** Always check your solution by plugging values back into the original problem. This helps confirm accuracy.

A person needs to arrange desks in a classroom such that the number of rows is the same as the number of columns of desks. After having made such an arrangement, he found that 34 desks were still left with him. When he doubled the number of rows and columns, he found that he was short of 14 desks. Find the total number of desks available with him.

Learn how to solve a quadratic system where a person arranges desks in a classroom. The number of desks left and shortage when doubling the rows and columns provide a system of equations. This detailed solution breaks down the process for high school students in Grades 9-11.

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