Divide 57 into two parts whose product is 782.

Answer

To solve the problem of dividing 57 into two parts whose product is 782, we can follow these steps:

### Step 1: Define the Variables
Let the two parts be \( x \) and \( y \).

### Step 2: Set Up the Equations
According to the problem, we have two equations:
1. The sum of the two parts:
\[ x + y = 57 \]
2. The product of the two parts:
\[ x \cdot y = 782 \]

### Step 3: Express \( y \) in Terms of \( x \)
From the first equation, we can express \( y \) in terms of \( x \):
\[ y = 57 - x \]

### Step 4: Substitute \( y \) in the Product Equation
Now, substitute \( y \) in the product equation:
\[ x \cdot (57 - x) = 782 \]

### Step 5: Expand the Equation
Expanding the left side gives:
\[ 57x - x^2 = 782 \]

### Step 6: Rearrange the Equation
Rearranging the equation to set it to zero:
\[ -x^2 + 57x - 782 = 0 \]
Multiplying through by -1 gives:
\[ x^2 - 57x + 782 = 0 \]

### Step 7: Factor the Quadratic Equation
Next, we need to factor the quadratic equation. We look for two numbers that multiply to 782 and add up to -57. The factors of 782 are 34 and 23:
\[ (x - 34)(x - 23) = 0 \]

### Step 8: Solve for \( x \)
Setting each factor to zero gives us:
1. \( x - 34 = 0 \) → \( x = 34 \)
2. \( x - 23 = 0 \) → \( x = 23 \)

### Step 9: Find Corresponding Values of \( y \)
Now, we can find the corresponding values of \( y \) using \( y = 57 - x \):
1. If \( x = 34 \):
\[ y = 57 - 34 = 23 \]
2. If \( x = 23 \):
\[ y = 57 - 23 = 34 \]

### Conclusion
The two parts are 34 and 23.

---