Multiple choice

If sin(α) and cos(α) are the roots of the equation ax2+ bx + c = 0, then b2 is

  1. c2 + 2ac

  2. a2 + ac

  3. a2 + 2ac

  4. c2 + ac
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Roots are sin(a) and cos(a). Sum = sin(a) + cos(a) = -b/a. Product = sin(a)cos(a) = c/a. Squaring the sum: sin^2(a) + cos^2(a) + 2sin(a)cos(a) = b^2/a^2. 1 + 2(c/a) = b^2/a^2. 1 + 2c/a = b^2/a^2. Multiply by a^2: a^2 + 2ac = b^2.

AI explanation

For the equation ax2 + bx + c = 0 with roots sin(alpha) and cos(alpha), the sum of roots gives sin(alpha) + cos(alpha) = -b/a. Squaring both sides yields sin2(alpha) + cos2(alpha) + 2 sin(alpha) cos(alpha) = b2/a2. Using the trigonometric identity sin2(alpha) + cos2(alpha) = 1, this becomes 1 + 2 sin(alpha) cos(alpha) = b2/a2. The product of the roots gives sin(alpha) cos(alpha) = c/a, so substituting this gives 1 + 2c/a = b2/a2. Multiplying by a2 results in a2 + 2ac = b2.

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