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If the difference between the roots of the equation x2 + kx + 1 = 0 is strictly less than √5, where |k| ≥ 2, then k can be any element of the interval
  • a)
    (−3, −2] ∪ [2, 3)
  • b)
    (−3, 3)
  • c)
    [−3, −2] ∪ [2, 3]
  • d)
    None of the above
Correct answer is option 'A'. Can you explain this answer?
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To find the difference between the roots of the equation, we can use the quadratic formula. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the roots are given by:

x = (-b ± √(b^2 - 4ac)) / 2a

In this case, our equation is x^2 + kx + 1 = 0, so a = 1, b = k, and c = 1. Substituting these values into the quadratic formula, we have:

x = (-k ± √(k^2 - 4(1)(1))) / 2(1)
= (-k ± √(k^2 - 4)) / 2

The difference between the roots is given by the absolute value of the difference between the two values of x:

Difference = |(-k + √(k^2 - 4)) / 2 - (-k - √(k^2 - 4)) / 2|
= |√(k^2 - 4) / 2 + √(k^2 - 4) / 2|
= |√(k^2 - 4) / 2 + √(k^2 - 4) / 2|
= |√(k^2 - 4) / 2|

To make the difference strictly less than 1, we want:

|√(k^2 - 4) / 2| < />

Squaring both sides of the inequality, we get:

(k^2 - 4) / 4 < />
k^2 - 4 < />
k^2 < />
-√8 < k="" />< />
-2√2 < k="" />< />

Therefore, the value of k must be between approximately -2.828 and 2.828 for the difference between the roots to be strictly less than 1.
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If the difference between the roots of the equation x2 + kx + 1 = 0 is strictly less than √5, where |k| ≥ 2, then kcan be any element of the intervala)(−3, −2] ∪ [2, 3)b)(−3, 3)c)[−3, −2] ∪ [2, 3]d)None of the aboveCorrect answer is option 'A'. Can you explain this answer?
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