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Consider the equation x2 + 2x – n = 0, where n Î N and n Î [5, 100]. Total number of different values of `n' so that the given equation has integral roots, is
- a)4
- b)6
- c)8
- d)3
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Consider the equation x2+ 2x –n = 0, where n Î N and n &Ic...
Root will be an integer when the “square root of (1+n)” must be a perfect square.
Between the numbers 5 to 100, perfect squares are 9,16,25,36,49,64,81,100
So, the values of n are 8,15,24,35,48,63,80,99 (so that n+1 is a perfect square)
Therefore, there are 8 different values of n between 5 and 100 so that the roots are integers.
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Between the numbers 5 to 100, perfect squares are 9,16,25,36,49,64,81,100
So, the values of n are 8,15,24,35,48,63,80,99 (so that n+1 is a perfect square)
Therefore, there are 8 different values of n between 5 and 100 so that the roots are integers.
Most Upvoted Answer
Consider the equation x2+ 2x –n = 0, where n Î N and n &Ic...
Approach:
To find the number of different values of n for which the given equation has integral roots, we need to consider the discriminant of the quadratic equation.
Discriminant:
The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by D = b^2 - 4ac.
Integral Roots:
For a quadratic equation to have integral roots, the discriminant should be a perfect square.
Solving the Equation:
Given equation: x^2 + 2x - n = 0
a = 1, b = 2, c = -n
Discriminant, D = 2^2 - 4(1)(-n) = 4 + 4n
Perfect Square:
For D to be a perfect square, 4 + 4n = k^2, where k is a natural number.
Values of n:
Now, we need to find the values of n such that 4 + 4n is a perfect square in the range [5, 100].
Calculation:
For n = 1, 4 + 4n = 8, not a perfect square.
For n = 2, 4 + 4n = 12, not a perfect square.
For n = 3, 4 + 4n = 16, a perfect square.
For n = 4, 4 + 4n = 20, not a perfect square.
For n = 5, 4 + 4n = 24, not a perfect square.
For n = 6, 4 + 4n = 28, not a perfect square.
For n = 7, 4 + 4n = 32, a perfect square.
For n = 8, 4 + 4n = 36, a perfect square.
For n = 9, 4 + 4n = 40, not a perfect square.
For n = 10, 4 + 4n = 44, not a perfect square.
For n = 11, 4 + 4n = 48, not a perfect square.
For n = 12, 4 + 4n = 52, not a perfect square.
For n = 13, 4 + 4n = 56, not a perfect square.
For n = 14, 4 + 4n = 60, not a perfect square.
For n = 15, 4 + 4n = 64, a perfect square.
For n = 16, 4 + 4n = 68, not a perfect square.
For n = 17, 4 + 4n = 72, not a perfect square.
For n = 18, 4 + 4n = 76, not a perfect square.
For n = 19, 4 + 4n = 80, not a perfect square.
For n = 20, 4 + 4n = 84, not a perfect square.
For n = 21, 4 + 4n = 88, not a perfect square.
For n = 22, 4 + 4n = 92, not
To find the number of different values of n for which the given equation has integral roots, we need to consider the discriminant of the quadratic equation.
Discriminant:
The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by D = b^2 - 4ac.
Integral Roots:
For a quadratic equation to have integral roots, the discriminant should be a perfect square.
Solving the Equation:
Given equation: x^2 + 2x - n = 0
a = 1, b = 2, c = -n
Discriminant, D = 2^2 - 4(1)(-n) = 4 + 4n
Perfect Square:
For D to be a perfect square, 4 + 4n = k^2, where k is a natural number.
Values of n:
Now, we need to find the values of n such that 4 + 4n is a perfect square in the range [5, 100].
Calculation:
For n = 1, 4 + 4n = 8, not a perfect square.
For n = 2, 4 + 4n = 12, not a perfect square.
For n = 3, 4 + 4n = 16, a perfect square.
For n = 4, 4 + 4n = 20, not a perfect square.
For n = 5, 4 + 4n = 24, not a perfect square.
For n = 6, 4 + 4n = 28, not a perfect square.
For n = 7, 4 + 4n = 32, a perfect square.
For n = 8, 4 + 4n = 36, a perfect square.
For n = 9, 4 + 4n = 40, not a perfect square.
For n = 10, 4 + 4n = 44, not a perfect square.
For n = 11, 4 + 4n = 48, not a perfect square.
For n = 12, 4 + 4n = 52, not a perfect square.
For n = 13, 4 + 4n = 56, not a perfect square.
For n = 14, 4 + 4n = 60, not a perfect square.
For n = 15, 4 + 4n = 64, a perfect square.
For n = 16, 4 + 4n = 68, not a perfect square.
For n = 17, 4 + 4n = 72, not a perfect square.
For n = 18, 4 + 4n = 76, not a perfect square.
For n = 19, 4 + 4n = 80, not a perfect square.
For n = 20, 4 + 4n = 84, not a perfect square.
For n = 21, 4 + 4n = 88, not a perfect square.
For n = 22, 4 + 4n = 92, not
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Consider the equation x2+ 2x –n = 0, where n Î N and n Î [5, 100]. Total number of different values of `n' so that the given equation has integral roots, isa)4b)6c)8d)3Correct answer is option 'C'. Can you explain this answer?
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Consider the equation x2+ 2x –n = 0, where n Î N and n Î [5, 100]. Total number of different values of `n' so that the given equation has integral roots, isa)4b)6c)8d)3Correct answer is option 'C'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Consider the equation x2+ 2x –n = 0, where n Î N and n Î [5, 100]. Total number of different values of `n' so that the given equation has integral roots, isa)4b)6c)8d)3Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the equation x2+ 2x –n = 0, where n Î N and n Î [5, 100]. Total number of different values of `n' so that the given equation has integral roots, isa)4b)6c)8d)3Correct answer is option 'C'. Can you explain this answer?.
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