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If p > 0, and x2 - 11x + p = 0 has integer roots, how many integer values can 'p' take?
- a)6
- b)11
- c)5
- d)10
- e)Infinitely many
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
If p > 0, and x2- 11x + p = 0 has integer roots, how many integer v...
Step 1: Compute sum and product of roots of this quadratic equation
The given quadratic equation is x2 - 11x + p = 0.
The sum of the roots of this quadratic equation =
The product of the roots of this quadratic equation =
The sum of the roots of this quadratic equation =
The product of the roots of this quadratic equation =
Step 2: Deduce properties about roots of this quadratic equation
Inference 1: The question states that p > 0.
'p' is the product of the roots of this quadratic equation. So, the product of the two roots is positive.
The product of two numbers is positive if either both the numbers are positive or both the numbers are negative.
'p' is the product of the roots of this quadratic equation. So, the product of the two roots is positive.
The product of two numbers is positive if either both the numbers are positive or both the numbers are negative.
Inference 2: We also know that the sum of the roots = 11, which is positive.
The sum of two negative numbers cannot be positive.
So, both the roots have to be positive.
The sum of two negative numbers cannot be positive.
So, both the roots have to be positive.
Step 3: List values possible for roots
We also know that the roots are integers.
So, we have to find different ways of expressing 11 as a sum of two positive integers.
Possibility 1: (1, 10)
Possibility 2: (2, 9)
Possibility 3: (3, 8)
Possibility 4: (4, 7)
Possibility 5: (5, 6)
So, we have to find different ways of expressing 11 as a sum of two positive integers.
Possibility 1: (1, 10)
Possibility 2: (2, 9)
Possibility 3: (3, 8)
Possibility 4: (4, 7)
Possibility 5: (5, 6)
Each of these pairs, will result in a different value for p.
So, p can take 5 different values.
So, p can take 5 different values.
The correct choice is (C) and the correct answer is 5.
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