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If both the roots of the quadratic equation x2 - mx + 4 = 0 are real and distinct and they lie in the interval [1 5] then m lies in the interval:
- a)(4,5)
- b)(3,4)
- c)(5,6)
- d)(–5,–4)
Correct answer is option 'A'. Can you explain this answer?
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If both the roots of the quadratic equation x2 - mx + 4 = 0 are real a...
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* If we consider αβ∈ (1,5) then option (1) is correct.
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If both the roots of the quadratic equation x2 - mx + 4 = 0 are real a...
2,3)
We know that the roots of the quadratic equation x2 - mx + 4 = 0 are real and distinct if the discriminant is greater than 0:
b2 - 4ac > 0
m2 - 4(1)(4) > 0
m2 > 16
m > 4 or m < />
Since we know that the roots lie in the interval [1,5], we can use the fact that the sum and product of the roots of a quadratic equation are related to the coefficients:
Sum of roots = -b/a = m/1 = m
Product of roots = c/a = 4/1 = 4
Let's call the roots r1 and r2. We know that:
1 ≤ r1 < r2="" ≤="" />
r1 + r2 = m
r1r2 = 4
We can use the quadratic formula to solve for the roots:
r1,2 = (m ± √(m2 - 16))/2
Since the roots are distinct, we know that r1 ≠ r2, which means that the discriminant is greater than 0:
m2 - 16 > 0
m2 > 16
m > 4 or m < />
We also know that the roots lie in the interval [1,5]:
1 ≤ r1 < r2="" ≤="" />
Using the quadratic formula, we can find the range of m that satisfies these conditions:
1 ≤ (m - √(m2 - 16))/2 < (m="" +="" √(m2="" -="" 16))/2="" ≤="" />
2 ≤ m - √(m2 - 16) < m="" +="" √(m2="" -="" 16)="" ≤="" />
4 ≤ 2√(m2 - 16) < 10="" -="" />
16 ≤ m2 - 16 < 100="" -="" 20m="" +="" />
0 ≤ 20m < />
0 ≤ m < />
Thus, the value of m lies in the interval (2,3). Therefore, the answer is (d).
We know that the roots of the quadratic equation x2 - mx + 4 = 0 are real and distinct if the discriminant is greater than 0:
b2 - 4ac > 0
m2 - 4(1)(4) > 0
m2 > 16
m > 4 or m < />
Since we know that the roots lie in the interval [1,5], we can use the fact that the sum and product of the roots of a quadratic equation are related to the coefficients:
Sum of roots = -b/a = m/1 = m
Product of roots = c/a = 4/1 = 4
Let's call the roots r1 and r2. We know that:
1 ≤ r1 < r2="" ≤="" />
r1 + r2 = m
r1r2 = 4
We can use the quadratic formula to solve for the roots:
r1,2 = (m ± √(m2 - 16))/2
Since the roots are distinct, we know that r1 ≠ r2, which means that the discriminant is greater than 0:
m2 - 16 > 0
m2 > 16
m > 4 or m < />
We also know that the roots lie in the interval [1,5]:
1 ≤ r1 < r2="" ≤="" />
Using the quadratic formula, we can find the range of m that satisfies these conditions:
1 ≤ (m - √(m2 - 16))/2 < (m="" +="" √(m2="" -="" 16))/2="" ≤="" />
2 ≤ m - √(m2 - 16) < m="" +="" √(m2="" -="" 16)="" ≤="" />
4 ≤ 2√(m2 - 16) < 10="" -="" />
16 ≤ m2 - 16 < 100="" -="" 20m="" +="" />
0 ≤ 20m < />
0 ≤ m < />
Thus, the value of m lies in the interval (2,3). Therefore, the answer is (d).
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If both the roots of the quadratic equation x2 - mx + 4 = 0 are real a...
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If both the roots of the quadratic equation x2 - mx + 4 = 0 are real and distinct and they lie in the interval [1 5] then m lies in the interval:a)(4,5)b)(3,4)c)(5,6) d)(–5,–4)Correct answer is option 'A'. Can you explain this answer?
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