Find two numbers whose sum is 27 and product is 182.
Numbers are 13 and 14.
Step 1: Assign variables. Let the two numbers be \(x\) and \(27 - x\), because their sum is given as 27.
Step 2: Use the product condition. Their product is 182, so we form the equation:
\[x(27 - x) = 182\]
Step 3: Expand and simplify.
\[27x - x^2 = 182\]
Rearrange to get the standard quadratic form:
\[-x^2 + 27x - 182 = 0\]
Multiply by \(-1\):
\[x^2 - 27x + 182 = 0\]
Step 4: Factorise the quadratic. We look for two numbers whose product is 182 and whose sum is 27. The pair is 13 and 14 since:
\[13 \times 14 = 182, \quad 13 + 14 = 27\]
So the factorisation is:
\[(x - 13)(x - 14) = 0\]
Step 5: Solve for \(x\).
\[x - 13 = 0 \Rightarrow x = 13\]
\[x - 14 = 0 \Rightarrow x = 14\]
Conclusion: The required numbers are 13 and 14.