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The smallest value of k, for which both the roots of the equation x2 - 8kx + 16(k2 - k + 1) = 0 are real, distinct and have values at least 4 , is
Correct answer is '2'. Can you explain this answer?
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The smallest value of k, for which both the roots of the equation x2 ...
The equation is x2 - 8kx + 16(k2 - k + 1) = 0
We have,
D = 64(k2 - (k2- k + 1)) = 64(k - 1) > 0,
⇒ k > 1
f(4) ≥ 0,
⇒ 16 - 32k + 16(k2 - k + 1)) ≥ 0
Which yield: k2 - 3k + 2 ≥ 0
⇒ k ≤ 1 or k ≥ 2.
Hence, k = 2
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The smallest value of k, for which both the roots of the equation x2 ...
Given equation: x^2 - 8kx + 16(k^2 - k + 1) = 0
To find the smallest value of k for which both roots of the equation are real, distinct, and have values at least 4, we need to analyze the discriminant of the quadratic equation.
Discriminant: The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by Δ = b^2 - 4ac. It helps determine the nature of the roots of the quadratic equation.
If the discriminant is positive (Δ > 0), the roots are real and distinct.
If the discriminant is zero (Δ = 0), the roots are real and equal.
If the discriminant is negative (Δ < 0),="" the="" roots="" are="" />
Analysis:
Let's calculate the discriminant of the given equation and set it greater than zero to ensure real and distinct roots.
Δ = (-8k)^2 - 4(1)(16(k^2 - k + 1))
= 64k^2 - 64(k^2 - k + 1)
= 64k^2 - 64k^2 + 64k - 64
= 64k - 64
We want the roots to be real and distinct, so Δ > 0.
64k - 64 > 0
64k > 64
k > 1
Thus, the value of k must be greater than 1 for the roots to be real and distinct.
Minimum value of k for roots at least 4:
To ensure that both roots have values at least 4, we can set up the following conditions:
Root 1: x ≥ 4
Root 2: x ≥ 4
Since the quadratic equation is symmetric, the sum of the roots is given by:
Sum of roots = -b/a = 8k/1 = 8k
For real and distinct roots, the sum of the roots must be greater than or equal to 8.
8k ≥ 8
k ≥ 1
Combining the conditions k > 1 and k ≥ 1, we find that the smallest value of k that satisfies both conditions is k = 2.
Hence, the correct answer is '2'.
To find the smallest value of k for which both roots of the equation are real, distinct, and have values at least 4, we need to analyze the discriminant of the quadratic equation.
Discriminant: The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by Δ = b^2 - 4ac. It helps determine the nature of the roots of the quadratic equation.
If the discriminant is positive (Δ > 0), the roots are real and distinct.
If the discriminant is zero (Δ = 0), the roots are real and equal.
If the discriminant is negative (Δ < 0),="" the="" roots="" are="" />
Analysis:
Let's calculate the discriminant of the given equation and set it greater than zero to ensure real and distinct roots.
Δ = (-8k)^2 - 4(1)(16(k^2 - k + 1))
= 64k^2 - 64(k^2 - k + 1)
= 64k^2 - 64k^2 + 64k - 64
= 64k - 64
We want the roots to be real and distinct, so Δ > 0.
64k - 64 > 0
64k > 64
k > 1
Thus, the value of k must be greater than 1 for the roots to be real and distinct.
Minimum value of k for roots at least 4:
To ensure that both roots have values at least 4, we can set up the following conditions:
Root 1: x ≥ 4
Root 2: x ≥ 4
Since the quadratic equation is symmetric, the sum of the roots is given by:
Sum of roots = -b/a = 8k/1 = 8k
For real and distinct roots, the sum of the roots must be greater than or equal to 8.
8k ≥ 8
k ≥ 1
Combining the conditions k > 1 and k ≥ 1, we find that the smallest value of k that satisfies both conditions is k = 2.
Hence, the correct answer is '2'.
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