Multiple choice

The equation $(1 + n^{2})x^{2} + 2ncx + (c^{2} - a^{2}) = 0$ will have equal roots if

  1. $c^{2} = 1 + a^{2}$
  2. $c^{2} = 1 - a^{2}$
  3. $c^{2} = 1 + n^{2} + a^{2}$
  4. $c^{2} = (1 + n^{2}) a^{2}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For equal roots, discriminant D = b^2 - 4ac = 0. (2nc)^2 - 4(1+n^2)(c^2-a^2) = 0. 4n^2c^2 - 4(c^2 - a^2 + n^2c^2 - n^2a^2) = 0. n^2c^2 - c^2 + a^2 - n^2c^2 + n^2a^2 = 0. a^2(1+n^2) = c^2.

AI explanation

For a quadratic equation to have equal roots, its discriminant must be zero, so we calculate D = B^2 - 4AC. Substituting the coefficients gives D = (2nc)^2 - 4(1 + n^2)(c^2 - a^2) = 0. Simplifying this yields 4n^2c^2 - 4(c^2 + n^2c^2 - a^2 - n^2a^2) = 0, which reduces to a^2(1 + n^2) - c^2 = 0. Solving for c^2, we get c^2 = (1 + n^2)a^2.

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