Multiple choice

If the quadratic equation x2 - px + q = 0 has roots that are squares of the roots of x2 + rx + s = 0, find the relation between p, r, and s.

  1. p = r2 - 4s

  2. p = r2 - 2s

  3. q = s2 - 4r

  4. p = s2 - r
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B Correct answer
Explanation

Let roots of x^2 + rx + s = 0 be a, b. Then a+b = -r and ab = s. Roots of x^2 - px + q = 0 are a^2, b^2. Then a^2+b^2 = p and a^2b^2 = q. Since a^2+b^2 = (a+b)^2 - 2ab, we have p = (-r)^2 - 2s = r^2 - 2s.

AI explanation

Let the roots of the equation x^2 + rx + s = 0 be u and v, which gives us the sum u + v = -r and the product uv = s. The roots of the new equation x^2 - px + q = 0 are u^2 and v^2, so the sum of these roots gives p = u^2 + v^2. We use the algebraic identity u^2 + v^2 = (u + v)^2 - 2uv to express p in terms of the given variables. Substituting the known values yields p = (-r)^2 - 2s, which simplifies to p = r^2 - 2s.

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