Which of the following is correct for the quadratic equation x2 + 2(a –1) x + a + 5 = 0
Text Solution
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(a, b, c)
D = 4(a – 1) 2 – 4(a + 5) = 4(a + 1) (a – 4)
Roots are real if D ≥ 0
⇒ (a + 1) (a – 4) ≥ 0
a ≤ – 1 or a ≥ 4
a ∈ (− ∞ , – 1] ∪ [4, ∞)
Roots are real if D ≥ 0, a + 5 > 0 and a – 1 < 0
coefficients of x 2 is 1 > 0
⇒ (a + 1) (a – 4) ≥ 0, a + 5 > 0 and a – 1 < 0
a ≤ – 1 or a ≥ 4 and a > – 5 and a < 1
⇒ a ∈ (– 5, – 1)
Roots are of opposite sign
D > 0 and a + 5 < 0 ⇒ a < – 1 or a > 4 and a < – 5
a < – 5 ⇒ a ∈ (– ∞ , – 5)
Roots are negative
if D ≥ 0, a + 5 > 0 and a – 1 > 0 ⇒ a ≤ – 1 or a ≥ 4
a > – 5 and a > 1 ⇒ a ≥ 4 ⇒ a ∈ [4, ∞ )
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