Multiple choice

The value of $\displaystyle a$ for which the equation $\displaystyle \left ( a^{2} - 3a + 2 \right )x^{2} + \left ( a^{2} - 5a + 4 \right )x + \left ( a^{2} - 1 \right ) = 0$ has more than two roots is:

  1. $\displaystyle -1$
  2. $\displaystyle 4$
  3. $\displaystyle 2$
  4. $\displaystyle 1$
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D Correct answer
Explanation

For a quadratic to have more than two roots, it must be an identity (0=0). This requires coefficients of x^2, x, and the constant term to be zero. a^2-1=0 implies a=1 or a=-1. If a=1, the equation becomes 0x^2 + 0x + 0 = 0, which is true for all x.

AI explanation

A quadratic equation can only have more than two roots if it simplifies to the trivial equation 0 = 0, which requires the coefficients of x^2, x, and the constant term to all simultaneously equal zero. Setting up the system a^2 - 3a + 2 = 0, a^2 - 5a + 4 = 0, and a^2 - 1 = 0, we find their respective roots are a = 1, 2, a = 1, 4, and a = 1, -1. The only common value of a that satisfies all three equations and reduces the expression to zero is 1.

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