Multiple choice

State whether the statement is true/false. If the roots of the equation $\left( {{a^2} + {b^2}} \right){x^2} - 2\left( {ac + bd} \right)x + \left( {{c^2} + {d^2}} \right) = 0$ are equal, then $\dfrac{a}{b} = \dfrac{c}{d}$.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For equal roots, the discriminant must be zero: 4(ac+bd)^2 - 4(a^2+b^2)(c^2+d^2) = 0. Expanding gives (ac)^2 + (bd)^2 + 2abcd - (a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2) = 0, which simplifies to -(ad-bc)^2 = 0. Thus ad = bc, or a/b = c/d.

AI explanation

For the equation to have equal roots, its discriminant must equal zero, so we set [ -2(ac + bd) ]^2 - 4(a^2 + b^2)(c^2 + d^2) = 0. Expanding this gives 4(a^2c^2 + 2abcd + b^2d^2) - 4(a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2) = 0. Simplifying the terms results in 8abcd - 4a^2d^2 - 4b^2c^2 = 0, which factors as -4(ad - bc)^2 = 0. This implies ad = bc, and by rearranging the terms, we get a/b = c/d, making the statement true.

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