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In the quadratic equation x2 + ax + b = 0, a and b can take any value from the set {1, 2, 3, 4}. How many pairs of values of a and b are possible in order that the quadratic equation has real roots?
- a)6
- b)7
- c)8
- d)16
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
In the quadratic equation x2+ ax + b = 0, a and b can take any value f...
x2 + ax + b = 0
For real roots, necessary condition is:
a2 – 4b ≥ 0
⇒ a2 ≥ 4b
And (a, b) ϵϵ {1, 2, 3, 4}
Possible pairs of a & b according to the above conditions:
(2, 1), (3, 1), (4, 1), (3, 2), (4, 2), (4, 3), (4, 4)
∴ There are 7 possible pairs of a and b.
For real roots, necessary condition is:
a2 – 4b ≥ 0
⇒ a2 ≥ 4b
And (a, b) ϵϵ {1, 2, 3, 4}
Possible pairs of a & b according to the above conditions:
(2, 1), (3, 1), (4, 1), (3, 2), (4, 2), (4, 3), (4, 4)
∴ There are 7 possible pairs of a and b.
Most Upvoted Answer
In the quadratic equation x2+ ax + b = 0, a and b can take any value f...
To determine the number of pairs of values of a and b that result in real roots for the quadratic equation, we need to analyze the discriminant of the equation. The discriminant is given by the formula: Δ = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.
Real roots exist when the discriminant is greater than or equal to zero, as this indicates that the quadratic equation intersects the x-axis.
Let's consider the given quadratic equation x^2 + ax + b = 0, where a and b can take any value from the set {1, 2, 3, 4}.
We can analyze the possible values of a and b in order to determine the number of pairs that satisfy the condition of real roots.
1. When a = 1:
- If b = 1, the discriminant is 1 - 4(1)(1) = -3, which is less than zero. So, there are no real roots.
- If b = 2, the discriminant is 4 - 4(1)(2) = -4, which is less than zero. So, there are no real roots.
- If b = 3, the discriminant is 9 - 4(1)(3) = -3, which is less than zero. So, there are no real roots.
- If b = 4, the discriminant is 16 - 4(1)(4) = 0, which is equal to zero. So, there is one real root.
2. When a = 2:
- If b = 1, the discriminant is 1 - 4(2)(1) = -7, which is less than zero. So, there are no real roots.
- If b = 2, the discriminant is 4 - 4(2)(2) = -12, which is less than zero. So, there are no real roots.
- If b = 3, the discriminant is 9 - 4(2)(3) = -15, which is less than zero. So, there are no real roots.
- If b = 4, the discriminant is 16 - 4(2)(4) = 0, which is equal to zero. So, there is one real root.
3. When a = 3:
- If b = 1, the discriminant is 1 - 4(3)(1) = -11, which is less than zero. So, there are no real roots.
- If b = 2, the discriminant is 4 - 4(3)(2) = -20, which is less than zero. So, there are no real roots.
- If b = 3, the discriminant is 9 - 4(3)(3) = -3, which is less than zero. So, there are no real roots.
- If b = 4, the discriminant is 16 - 4(3)(4) = 16, which is greater than zero. So, there are two real roots.
4. When a = 4:
- If b = 1, the discriminant is 1 -
Real roots exist when the discriminant is greater than or equal to zero, as this indicates that the quadratic equation intersects the x-axis.
Let's consider the given quadratic equation x^2 + ax + b = 0, where a and b can take any value from the set {1, 2, 3, 4}.
We can analyze the possible values of a and b in order to determine the number of pairs that satisfy the condition of real roots.
1. When a = 1:
- If b = 1, the discriminant is 1 - 4(1)(1) = -3, which is less than zero. So, there are no real roots.
- If b = 2, the discriminant is 4 - 4(1)(2) = -4, which is less than zero. So, there are no real roots.
- If b = 3, the discriminant is 9 - 4(1)(3) = -3, which is less than zero. So, there are no real roots.
- If b = 4, the discriminant is 16 - 4(1)(4) = 0, which is equal to zero. So, there is one real root.
2. When a = 2:
- If b = 1, the discriminant is 1 - 4(2)(1) = -7, which is less than zero. So, there are no real roots.
- If b = 2, the discriminant is 4 - 4(2)(2) = -12, which is less than zero. So, there are no real roots.
- If b = 3, the discriminant is 9 - 4(2)(3) = -15, which is less than zero. So, there are no real roots.
- If b = 4, the discriminant is 16 - 4(2)(4) = 0, which is equal to zero. So, there is one real root.
3. When a = 3:
- If b = 1, the discriminant is 1 - 4(3)(1) = -11, which is less than zero. So, there are no real roots.
- If b = 2, the discriminant is 4 - 4(3)(2) = -20, which is less than zero. So, there are no real roots.
- If b = 3, the discriminant is 9 - 4(3)(3) = -3, which is less than zero. So, there are no real roots.
- If b = 4, the discriminant is 16 - 4(3)(4) = 16, which is greater than zero. So, there are two real roots.
4. When a = 4:
- If b = 1, the discriminant is 1 -
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In the quadratic equation x2+ ax + b = 0, a and b can take any value from the set {1, 2, 3, 4}. How many pairs of values of a and b are possible in order that the quadratic equation has real roots?a)6b)7c)8d)16Correct answer is option 'B'. Can you explain this answer?
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In the quadratic equation x2+ ax + b = 0, a and b can take any value from the set {1, 2, 3, 4}. How many pairs of values of a and b are possible in order that the quadratic equation has real roots?a)6b)7c)8d)16Correct answer is option 'B'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about In the quadratic equation x2+ ax + b = 0, a and b can take any value from the set {1, 2, 3, 4}. How many pairs of values of a and b are possible in order that the quadratic equation has real roots?a)6b)7c)8d)16Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In the quadratic equation x2+ ax + b = 0, a and b can take any value from the set {1, 2, 3, 4}. How many pairs of values of a and b are possible in order that the quadratic equation has real roots?a)6b)7c)8d)16Correct answer is option 'B'. Can you explain this answer?.
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