A two digit number is such that the product, of the gitis is 24. When 18 is added to the number, the digits intercharge their places. Formulate the quadratic equation whose roots are the digits of the number.

Answer

To solve the problem step by step, we will follow the logic presented in the video transcript while ensuring clarity in each step.

### Step 1: Define the Digits
Let the two-digit number be represented as \(10x + y\), where \(x\) is the tens digit and \(y\) is the units digit.

### Step 2: Set Up the Product of the Digits
According to the problem, the product of the digits is given as:
\[ xy = 24 \]
From this, we can express \(y\) in terms of \(x\):
\[ y = \frac{24}{x} \]

### Step 3: Set Up the Equation for Interchanging Digits
We know that when 18 is added to the number, the digits interchange their places. This gives us the equation:
\[ 10x + y + 18 = 10y + x \]

### Step 4: Substitute \(y\) in the Equation
Substituting \(y = \frac{24}{x}\) into the equation:
\[ 10x + \frac{24}{x} + 18 = 10 \left(\frac{24}{x}\right) + x \]

### Step 5: Simplify the Equation
Now, simplify both sides:
\[ 10x + \frac{24}{x} + 18 = \frac{240}{x} + x \]
Multiply through by \(x\) to eliminate the fraction:
\[ 10x^2 + 24 + 18x = 240 + x^2 \]

### Step 6: Rearrange the Equation
Rearranging gives:
\[ 10x^2 - x^2 + 18x + 24 - 240 = 0 \]
This simplifies to:
\[ 9x^2 + 18x - 216 = 0 \]

### Step 7: Divide the Equation by 9
To simplify further, divide the entire equation by 9:
\[ x^2 + 2x - 24 = 0 \]

### Final Result
The quadratic equation whose roots are the digits of the number is:
\[ x^2 + 2x - 24 = 0 \]