If r is a constant such that |x2 – 4x - 13| = r has exactly three distinct real roots, then the value of r is?
Explanation:
Given, |x2 – 4x - 13| = r ∴ x2 – 4x - 13 = ± r
We have two quadratic equations here but only three distinct roots it means one of the quadratic equations will have equal roots.
Case 1: x2 – 4x - 13 = r has equal roots, i.e., Discriminant = 0 ⇒ x2 – 4x – 13 - r = 0 had D = 0
⇒ D = 16 – 4(-13 - r) = 0 ⇒ 16 + 52 + 4r = 0 ⇒ r = - 17
Case 2: x2 – 4x - 13 = - r has equal roots, i.e., Discriminant = 0 ⇒ x2 – 4x – 13 + r = 0 had D = 0
⇒ D = 16 – 4(-13 + r) = 0 ⇒ 16 + 52 - 4r = 0 ⇒ r = 17
Hence, option (c).
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