Multiple choice

What is the condition for one root of the quadratic equation $ax^2 + bx + c = 0$ to be twice the other?

  1. $b^2 = 4ac$
  2. $2b^2=9ac$
  3. $c^2 = 4a + b^2$
  4. $c^2 = 9a -b^2$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let roots be alpha and 2*alpha. Sum = 3*alpha = -b/a, Product = 2*alpha^2 = c/a. From sum, alpha = -b/(3a). Substitute into product: 2*(-b/(3a))^2 = c/a => 2*(b^2 / 9a^2) = c/a => 2b^2 / 9a = c => 2b^2 = 9ac.

AI explanation

Let the roots be y and 2y. Using Vieta's formulas, the sum of the roots is y + 2y = 3y = -b divided by a, so b = -3ay. The product of the roots is y * 2y = 2y^2 = c divided by a, so c = 2ay^2. Substituting y = -b divided by 3a into the product equation gives c = 2a(b^2 divided by 9a^2), which simplifies to 9ac = 2b^2.

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