Multiple choice

If the equation $(a^2+b^2)x^2-2(ac+bd)x+c^2+d^2=0$ has equal roots, then

  1. $ab=cd$
  2. $ad=bc$
  3. $ad=\sqrt{bc}$
  4. $ab=\sqrt{cd}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For equal roots, the discriminant D = 0. D = [2(ac+bd)]^2 - 4(a^2+b^2)(c^2+d^2) = 0. 4(a^2c^2 + 2abcd + b^2d^2) - 4(a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2) = 0. 2abcd - a^2d^2 - b^2c^2 = 0. -(ad - bc)^2 = 0. Thus, ad = bc.

AI explanation

For the equation to have equal roots, its discriminant must be zero, so (-2(ac+bd))^2 - 4(a^2+b^2)(c^2+d^2) = 0. Expanding both terms gives 4(a^2c^2 + b^2d^2 + 2abcd) - 4(a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2) = 0. This simplifies to 8abcd - 4a^2d^2 - 4b^2c^2 = 0, which can be rearranged into the perfect square (ad - bc)^2 = 0. Therefore, ad = bc.

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