The sum of two numbers is 5; their product is -14. Find the numbers.

Analyze the question: We need two numbers whose sum is 5 and whose product is -14.

Concept: If two numbers have sum S and product P, they are the roots of the quadratic equation:

x² − (sum)x + (product) = 0

Solution:

x² − 5x − 14 = 0

We need two numbers that multiply to -14 and add to -5. These are -7 and +2.

x² − 7x + 2x − 14 = 0 x(x − 7) + 2(x − 7) = 0 (x − 7)(x + 2) = 0

So x = 7 or x = -2.

Check:

• Sum: 7 + (−2) = 5 ✓
• Product: 7 × (−2) = −14 ✓

Checking the other options:

• A: 5 + 1 = 6, and the product is 5, so no.
• B: 7 + 1 = 8, so no.
• C: 7 + 0 = 7, and the product is 0, so no.

Tip: In "sum and product" questions, you can test the options directly. Add the two numbers, then multiply them. Only one option will satisfy both conditions, and this saves time in the exam.