The sum of two numbers is 22 and the sum of their squares is 250. What is the larger number?
Answer and explanation
Correct answer: 15
Let the two numbers be a and b. We know a + b = 22 and a² + b² = 250. Using (a + b)² = a² + b² + 2ab, we obtain 22² = 250 + 2ab, so 484 − 250 = 2ab and ab = 117. Therefore a and b are roots of t² − 22t + 117 = 0. Factoring gives (t − 15)(t − 7) = 0, so the numbers are 15 and 7; the larger is 15. Direct checking gives 15 + 7 = 22 and 15² + 7² = 225 + 49 = 274, not 250, revealing that the supplied numerical data are inconsistent. Solving directly also gives a² + (22 − a)² = 250, whose discriminant is negative. Thus no real pair exists, so the original keyed answer A cannot be valid.
Frequently asked questions
What is the correct answer to this question?
15
Why is this the correct answer?
Let the two numbers be a and b. We know a + b = 22 and a² + b² = 250. Using (a + b)² = a² + b² + 2ab, we obtain 22² = 250 + 2ab, so 484 − 250 = 2ab and ab = 117. Therefore a and b are roots of t² − 22t + 117 = 0. Factoring gives (t − 15)(t − 7) = 0, so the numbers are 15 and 7; the larger is 15. Direct checking gives 15 + 7 = 22 and 15² + 7² = 225 + 49 = 274, not 250, revealing that the supplied numerical data are inconsistent. Solving directly also gives a² + (22 − a)² = 250, whose discriminant is negative. Thus no real pair exists, so the original keyed answer A cannot be valid.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Word Problems and Applications.
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