Question

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

• A. \( 13, 12 \)
• B. \( 13, 14 \)
• C. \( 15, 12 \)
• D. \( 11, 16 \)

Hint:

When looking for two numbers given their sum and product, consider forming a quadratic equation where the numbers are the roots. If the sum is \( S \) and the product is \( P \), the equation is \( t^2 - St + P = 0 \).

Answer 1

The Correct Option is B

Step 1: Set up the equations based on the given information.
Let the two numbers be \( x \) and \( y \). We are given: \[ x + y = 27 \quad \cdots (1) \] \[ xy = 182 \quad \cdots (2) \] Step 2: Solve the system of equations.
From equation (1), we can express \( y \) in terms of \( x \): \( y = 27 - x \). Substitute this into equation (2): \[ x(27 - x) = 182 \] \[ 27x - x^2 = 182 \] \[ x^2 - 27x + 182 = 0 \] Step 3: Solve the quadratic equation.
We can solve this quadratic equation by factoring. We need two numbers that multiply to 182 and add up to -27. These numbers are -13 and -14. \[ (x - 13)(x - 14) = 0 \] So, the possible values for \( x \) are \( x = 13 \) or \( x = 14 \). Step 4: Find the corresponding values of \( y \).
If \( x = 13 \), then from equation (1), \( y = 27 - 13 = 14 \).
If \( x = 14 \), then from equation (1), \( y = 27 - 14 = 13 \).
Thus, the two numbers are 13 and 14.