Multiple choice

About the equation of $ax^2 - 2x \sqrt2 + c =0$, with a and c real constants, we are told that the discriminant is zero. The roots are necessarily:

  1. equal and integral

  2. equal and rational

  3. equal and real

  4. equal and irrational

  5. equal and imaginary
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

If the discriminant of a quadratic equation with real coefficients is zero, the roots are equal and real. Since the coefficients are real, the roots must be real.

AI explanation

The discriminant of the quadratic equation ax^2 - 2x times the square root of 2 + c = 0 is calculated as b squared minus 4ac. Substituting the values gives (negative 2 times the square root of 2) squared minus 4ac, which simplifies to 8 minus 4ac. Since we are given that the discriminant is zero, the roots are given by the quadratic formula as (2 times the square root of 2) divided by 2a, meaning the roots are real numbers. Because the discriminant is zero, the two real roots are exactly equal to each other.

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