Multiple choice

If the ratio of the roots of the equation $x^2+bx+c=0$ is the same as that of $x^2 + qx +r=0$, then

  1. $r^2b = qc^2$
  2. $r^2c = qb^2$
  3. $c^2r = q^2b$
  4. $b^2r=q^2c$
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D Correct answer
Explanation

Let roots of the first be a, b. a/b = k. Roots of second are ka, kb. Sum of roots: a(1+k) = -b, a(1+k) = -q/k. Product: a^2 k = c, a^2 k^2 = r. Ratio of products: c/r = 1/k^2. k^2 = r/c. From sum: (a(1+k))^2 = b^2 => a^2(1+k)^2 = b^2. c/k * (1+k)^2 / k = b^2. Substituting k^2 = r/c leads to the relation b^2r = q^2c.

AI explanation

Let the roots of the first equation be ka and kb, making their sum k(a + b) = -b and their product k^2 ab = c. Let the identical ratio roots of the second equation be la and lb, so their sum l(a + b) = -q and their product l^2 ab = r. By expressing the ratio of the square of the sum to the product for both equations, we have (-b)^2/c = (-q)^2/r. Cross-multiplying this relationship gives b^2 r = q^2 c.

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