Multiple choice

If one root of the equation $x^2 - 30x + p = 0$ is square of the other, then $p$ is equal to

  1. only $125$
  2. $125, -216$
  3. $125, 215$
  4. only $216$
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B Correct answer
Explanation

Let roots be a and a^2. Sum of roots: a + a^2 = 30. Product of roots: a * a^2 = a^3 = p. From a^2 + a - 30 = 0, (a+6)(a-5) = 0. If a=5, a^3 = 125. If a=-6, a^3 = -216. Thus p = 125 or -216.

AI explanation

Let the roots of the quadratic equation be a and a squared. The product of the roots equals p, so a cubed equals p. The sum of the roots is 30, giving a plus a squared equals 30, or a squared plus a minus 30 equals 0. Factoring this gives (a minus 5)(a plus 6) equals 0, meaning a is 5 or negative 6. If a is 5, then p is 125, and if a is negative 6, then p is negative 216. The possible values for p are 125 and negative 216.

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