Solve the following equations
(i) x 2 – 7|x| – 8 = 0
(ii) |x 2 – x + 1| = |x 2 – x – 1|
(iii) |x 3 – 6x 2 + 11x – 6| = 6
(iv) |x 2 – 2x| + x = 6
(v) |x 2 – x – 6| = x + 2.
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
(i) |x| 2 – 7|x| – 8 = 0 ⇒ (|x| – 8)(|x| + 1) = 0 ⇒ |x| = 8 ⇒ x = ±8
(ii) Squaring both sides, we get
(x 2 – x + 1) 2 – (x 2 – x – 1) 2 = 0 ⇒ (2x 2 – 2x)(2) = 0 ⇒ x = 0, 1
(iii) |x 3 – 6x 2 + 11x – 6| = 6
x 3 – 6x 2 + 11x – 6 = 6 ⇒ x 3 – 6x 2 + 11x – 12 = 0
⇒ (x – 4)(x 2 – 2x + 6) = 0 ⇒ x = 4
or x 3 – 6x 2 + 11x – 6 = –6 ⇒ x(x 2 – 6x + 11) = 0 ⇒ x = 0
(iv) Case -I : x ∈ (– ∞ , 0] ∪ [2, ∞ )
x 2 – 2x + x = 6 ⇒ x 2 – x – 6 = 0 ⇒ x = –2, 3
Case-II : x ∈ [0, 2]
2x – x 2 + x = 6 ⇒ x 2 – 3x + 6 = 0 ⇒ No real roots
(v) Case-I : x ∈ [–2, 3]
6 + x – x 2 = x + 2 ⇒ x 2 = 4 ⇒ x = ±2
Case-II : x ∈ (– ∞ , –2] ∪ [3, ∞ )
x 2 – x – 6 = x + 2 ⇒ x 2 – 2x – 8 = 0 ⇒ (x – 4)(x + 2) = 0
⇒ x = –2, 4
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