3
advanced-math

Solve for xx: (x2−1)2−5(x2−1)+6=0(x^2 - 1)^2 - 5(x^2 - 1) + 6 = 0.

A

±2,±3\pm 2, \pm \sqrt{3}

B

±1,±2\pm 1, \pm 2

C

2,32, 3

D

±2,±3\pm \sqrt{2}, \pm \sqrt{3}

Correct Answer: A

Choice A is the correct answer. This is quadratic in form. Let u=x2−1u = x^2 - 1.

  1. u2−5u+6=0u^2 - 5u + 6 = 0.
  2. (u−2)(u−3)=0⇒u=2(u - 2)(u - 3) = 0 \Rightarrow u = 2 or u=3u = 3.
  3. Case 1: x2−1=2⇒x2=3⇒x=±3x^2 - 1 = 2 \Rightarrow x^2 = 3 \Rightarrow x = \pm \sqrt{3}.
  4. Case 2: x2−1=3⇒x2=4⇒x=±2x^2 - 1 = 3 \Rightarrow x^2 = 4 \Rightarrow x = \pm 2.

Choice B is incorrect. Choice C is incorrect. Choice D is incorrect.

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Set 20: Quadratic Equations and Functions 20