Two positive numbers differ by 9, and the sum of their squares is 585. What is the smaller number?
Answer and explanation
Correct answer: 12
Let the smaller number be \(x\); then the larger number is \(x+9\). Thus, \(x^2+(x+9)^2=585\), which simplifies to \(2x^2+18x-504=0\) and then \(x^2+9x-252=0\). Factoring gives \((x+21)(x-12)=0\), so \(x=12\) or \(x=-21\). Since both numbers are positive, \(x=12\) is valid. In the exam, reject any root that violates the stated positivity condition.
Frequently asked questions
What is the correct answer to this question?
12
Why is this the correct answer?
Let the smaller number be \(x\); then the larger number is \(x+9\). Thus, \(x^2+(x+9)^2=585\), which simplifies to \(2x^2+18x-504=0\) and then \(x^2+9x-252=0\). Factoring gives \((x+21)(x-12)=0\), so \(x=12\) or \(x=-21\). Since both numbers are positive, \(x=12\) is valid. In the exam, reject any root that violates the stated positivity condition.
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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