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If a ∈ Z and the equation (x - a)(x -10) +1 = 0 has integral roots, then values of ‘a’ are
- a)10, 8
- b)12, 10
- c)12, 8
- d)10, 12
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
If a ∈ Z and the equation (x - a)(x -10) +1 = 0 has integral roo...
(x – a)(x – 10) = -1
Since a is an integer and x is also integer,
So, (x – a) & (x – 10) can be 1 or –1
For (x – 10) = 1, x = 11
(x – a) = -1, 11 – a = -1, a = 12
For (x – 10) = -1, x = 9
(x – a) = 1, 9 – a = 1, a = 8
Since a is an integer and x is also integer,
So, (x – a) & (x – 10) can be 1 or –1
For (x – 10) = 1, x = 11
(x – a) = -1, 11 – a = -1, a = 12
For (x – 10) = -1, x = 9
(x – a) = 1, 9 – a = 1, a = 8
Most Upvoted Answer
If a ∈ Z and the equation (x - a)(x -10) +1 = 0 has integral roo...
Given equation: (x - a)(x - 10) = 1 = 0
To find the values of 'a' for which the equation has integral roots, we need to solve the equation and determine the conditions for the roots to be integers.
Solving the equation:
Expanding the equation, we get:
x^2 - (10 + a)x + 10a - 1 = 0
Since the equation has integral roots, the discriminant (b^2 - 4ac) must be a perfect square.
Finding the discriminant:
The discriminant of the equation is:
D = (10 + a)^2 - 4(10a - 1)
= 100 + 20a + a^2 - 40a + 4
= a^2 - 20a + 104
To ensure that the discriminant is a perfect square, let's rewrite it as:
D = (a - 10)^2 + 4(26)
Conditions for integral roots:
For the discriminant to be a perfect square, the term (a - 10)^2 + 4(26) must be a perfect square.
Let's analyze the possible values of (a - 10)^2 + 4(26):
1. If (a - 10)^2 + 4(26) = 4k^2, where k is an integer, then the equation has integral roots.
Checking the given options:
a) (10 - 10)^2 + 4(26) = 104
This is not a perfect square, so option 'A' is incorrect.
b) (12 - 10)^2 + 4(26) = 144
This is a perfect square, so option 'B' satisfies the condition.
c) (12 - 10)^2 + 4(26) = 144
This is a perfect square, so option 'C' satisfies the condition.
d) (10 - 10)^2 + 4(26) = 104
This is not a perfect square, so option 'D' is incorrect.
Conclusion:
The equation (x - a)(x - 10) = 1 = 0 has integral roots for the values of 'a' given in option 'B' (12, 10) and option 'C' (12, 8). However, since option 'B' is not listed as a correct answer, the correct option is 'C' (12, 8).
To find the values of 'a' for which the equation has integral roots, we need to solve the equation and determine the conditions for the roots to be integers.
Solving the equation:
Expanding the equation, we get:
x^2 - (10 + a)x + 10a - 1 = 0
Since the equation has integral roots, the discriminant (b^2 - 4ac) must be a perfect square.
Finding the discriminant:
The discriminant of the equation is:
D = (10 + a)^2 - 4(10a - 1)
= 100 + 20a + a^2 - 40a + 4
= a^2 - 20a + 104
To ensure that the discriminant is a perfect square, let's rewrite it as:
D = (a - 10)^2 + 4(26)
Conditions for integral roots:
For the discriminant to be a perfect square, the term (a - 10)^2 + 4(26) must be a perfect square.
Let's analyze the possible values of (a - 10)^2 + 4(26):
1. If (a - 10)^2 + 4(26) = 4k^2, where k is an integer, then the equation has integral roots.
Checking the given options:
a) (10 - 10)^2 + 4(26) = 104
This is not a perfect square, so option 'A' is incorrect.
b) (12 - 10)^2 + 4(26) = 144
This is a perfect square, so option 'B' satisfies the condition.
c) (12 - 10)^2 + 4(26) = 144
This is a perfect square, so option 'C' satisfies the condition.
d) (10 - 10)^2 + 4(26) = 104
This is not a perfect square, so option 'D' is incorrect.
Conclusion:
The equation (x - a)(x - 10) = 1 = 0 has integral roots for the values of 'a' given in option 'B' (12, 10) and option 'C' (12, 8). However, since option 'B' is not listed as a correct answer, the correct option is 'C' (12, 8).
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Community Answer
If a ∈ Z and the equation (x - a)(x -10) +1 = 0 has integral roo...
For integral roots, b²-4ac must be a perfect square.
=>(a-10)²=2
=>(a-10)²=2
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