Multiple choice

If $\displaystyle \alpha $ is a root repeated twice of the quadratic equation $\displaystyle \left ( a-d \right )x^{2}+ax+\left ( a+d \right )=0 $ then $\displaystyle \frac{d^{2}}{a^{2}}$ has the value equal to

  1. $\displaystyle \sin ^{2}90^{\circ}$
  2. $\displaystyle \cos ^{2}60^{\circ}$
  3. $\displaystyle \sin ^{2}45^{\circ}$
  4. $\displaystyle \cos ^{2}30^{\circ}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

If alpha is a repeated root, the discriminant must be zero. For (a-d)x^2 + ax + (a+d) = 0, D = a^2 - 4(a-d)(a+d) = a^2 - 4(a^2 - d^2) = 0. So a^2 - 4a^2 + 4d^2 = 0, which means 4d^2 = 3a^2, or d^2/a^2 = 3/4. This equals cos^2(30 degrees) = (sqrt(3)/2)^2 = 3/4.

AI explanation

Because the quadratic equation has a root repeated twice, its discriminant must equal zero, so we set a^2 - 4(a - d)(a + d) = 0. Expanding and simplifying this condition yields a^2 - 4(a^2 - d^2) = 0, which means 4d^2 = 3a^2. Dividing both sides by 4a^2 gives d^2/a^2 = 3/4, which corresponds exactly to the trigonometric value of cos^2(30 degrees).

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