The roots of x^2-px+36=0 are positive integers and their difference is 5. What is p?
Answer and explanation
Correct answer: 13
Let the positive integer roots be r and s, with r>s. Vieta’s product relation gives rs=36, and the stated difference gives r-s=5. The positive factor pairs of 36 are (1,36), (2,18), (3,12), and (4,9). Only 9-4=5, so the roots must be 9 and 4. Their sum is 9+4=13. For x^2-px+36=0, Vieta’s sum relation says r+s=p, because the coefficient of x is -p. Therefore p=13, making option C correct. The other options do not equal the sum of the only positive factor pair whose difference is 5. This also confirms that the quadratic is x^2-13x+36=(x-9)(x-4).
Frequently asked questions
What is the correct answer to this question?
13
Why is this the correct answer?
Let the positive integer roots be r and s, with r>s. Vieta’s product relation gives rs=36, and the stated difference gives r-s=5. The positive factor pairs of 36 are (1,36), (2,18), (3,12), and (4,9). Only 9-4=5, so the roots must be 9 and 4. Their sum is 9+4=13. For x^2-px+36=0, Vieta’s sum relation says r+s=p, because the coefficient of x is -p. Therefore p=13, making option C correct. The other options do not equal the sum of the only positive factor pair whose difference is 5. This also confirms that the quadratic is x^2-13x+36=(x-9)(x-4).
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Roots of a Quadratic Equation.
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