Multiple choice

If $\alpha \epsilon \left( -1,1 \right) $ then roots of the quadratic equation $\left( a-1 \right) { x }^{ 2 }+ax+\sqrt { 1-{ a }^{ 2 } } =0$ are

  1. real

  2. imaginary

  3. both equal

  4. none of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For -1 < a < 1, the coefficient a - 1 is negative and sqrt(1 - a^2) is positive. Thus the discriminant a^2 - 4(a - 1)sqrt(1 - a^2) is positive, so both roots are real.

AI explanation

The discriminant of the quadratic equation is given by D equals a squared minus 4 times the quantity (a minus 1) times the square root of (1 minus a squared). For alpha in the interval (negative 1, 1), the coefficient (a minus 1) is negative and the square root of (1 minus a squared) is positive, making the entire second term positive. Since a squared is also positive, adding these values yields a positive discriminant, proving the roots are real.

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