Find two such numbers whose sum is 27 and product is 182.

Answer

To find two numbers whose sum is 27 and product is 182, we can follow these steps:

### Step 1: Set up the equations
Let the two numbers be \( x \) and \( y \). According to the problem, we have:
1. \( x + y = 27 \) (Equation 1)
2. \( x \cdot y = 182 \) (Equation 2)

### Step 2: Express one variable in terms of the other
From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 27 - x \]

### Step 3: Substitute into the product equation
Now, substitute \( y \) in Equation 2:
\[ x \cdot (27 - x) = 182 \]

### Step 4: Expand and rearrange the equation
Expanding the left side gives:
\[ 27x - x^2 = 182 \]
Rearranging this equation, we get:
\[ x^2 - 27x + 182 = 0 \]

### Step 5: Solve the quadratic equation
Now, we will use the quadratic formula to solve for \( x \):
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -27 \), and \( c = 182 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-27)^2 - 4 \cdot 1 \cdot 182 = 729 - 728 = 1 \]

Now substituting into the quadratic formula:
\[ x = \frac{27 \pm \sqrt{1}}{2 \cdot 1} \]
\[ x = \frac{27 \pm 1}{2} \]

Calculating the two possible values for \( x \):
1. \( x = \frac{28}{2} = 14 \)
2. \( x = \frac{26}{2} = 13 \)

### Step 6: Find the corresponding values of \( y \)
Now, we can find the corresponding values of \( y \) using \( y = 27 - x \):
1. If \( x = 14 \), then \( y = 27 - 14 = 13 \).
2. If \( x = 13 \), then \( y = 27 - 13 = 14 \).

### Conclusion
The two numbers are \( 14 \) and \( 13 \).