Multiple choice

Solve the following quadratic equation using quadratic formula: $a(x^2+1)=x(a^2+1)$

  1. $\left \{ \dfrac{1}{a}, a \right \}$
  2. $\left \{ \dfrac{1}{a}, 1 \right \}$
  3. $\left \{ \dfrac{1}{a}, a^2 \right \}$

  4. None of these
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A Correct answer
Explanation

Expand the equation: ax^2 + a = a^2x + x. Rearrange to standard form: ax^2 - (a^2 + 1)x + a = 0. Using the quadratic formula or factoring: (ax - 1)(x - a) = 0. The roots are 1/a and a.

AI explanation

Rearrange the equation to standard form to get ax^2 - (a^2 + 1)x + a = 0. Using the quadratic formula, the discriminant is (a^2 + 1)^2 - 4(a)(a), which simplifies to a^4 - 2a^2 + 1 or (a^2 - 1)^2. Substituting this back into the formula gives x = ((a^2 + 1) +/- (a^2 - 1)) / 2a, yielding the solutions a and 1/a.

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