Find two numbers whose sum is 29 and whose product is 100.

Introduction / Context:
This is a classic quadratic-based word problem. We are told the sum and product of two numbers and asked to find the numbers themselves. Such questions map naturally to quadratic equations, since if the numbers are x and y, then x + y and xy correspond to the coefficients and constant term of a quadratic with roots x and y. Recognising this structure allows an efficient algebraic solution.

Given Data / Assumptions:

• There are two numbers, call them x and y.
• Their sum is x + y = 29.
• Their product is x * y = 100.
• We must find the actual values of x and y.
• Numbers are assumed to be real, and the options suggest positive integer solutions.

Concept / Approach:
If x and y are the roots of a quadratic equation, then the equation can be written as t^2 - (sum of roots) * t + (product of roots) = 0. Here, the sum of roots is 29 and the product is 100, so we construct the quadratic t^2 - 29t + 100 = 0. Solving this quadratic will directly give the two numbers. This approach is systematic and avoids guesswork, though we can later cross-check with the options by simple substitution.

Step-by-Step Solution:
Step 1: Let the two numbers be x and y.Step 2: We know x + y = 29 and x * y = 100.Step 3: Form the quadratic whose roots are x and y: t^2 - (x + y)t + x * y = 0.Step 4: Substitute the known sum and product: t^2 - 29t + 100 = 0.Step 5: Solve this quadratic using the discriminant method. The discriminant D is 29^2 - 4 * 1 * 100 = 841 - 400 = 441.Step 6: The square root of D is sqrt(441) = 21.Step 7: The roots are t = [29 ± 21] / 2.Step 8: Compute the two roots: t1 = (29 + 21) / 2 = 50 / 2 = 25 and t2 = (29 - 21) / 2 = 8 / 2 = 4.Step 9: Therefore, the two numbers are 25 and 4.

Verification / Alternative check:
Check the sum: 25 + 4 = 29, which matches the given sum. Check the product: 25 * 4 = 100, which matches the given product. Both conditions are satisfied, so the pair (25, 4) is correct. Also, the order does not matter since (4, 25) is the same pair of numbers in a different order.

Why Other Options Are Wrong:
Option 20 and 5: Sum is 25 and product is 100, so the sum condition fails.
Option 20 and 9: Sum is 29 but product is 180, not 100.
Option 10 and 10: Sum is 20 and product is 100, failing the sum condition.
Option 19 and 10: Sum is 29, but product is 190, not 100.

Common Pitfalls:
Some learners rely entirely on trial and error with the options and may not systematically check both sum and product. Others might form the quadratic equation incorrectly (for example, writing t^2 + 29t + 100). Another common mistake is miscomputing the discriminant or its square root. Using clear algebraic steps helps to avoid these errors.

Final Answer:
The required two numbers are 25 and 4.