Multiple choice

If $\alpha ,{ \alpha }^{ 2 }$ are the roots of the equation ${ x }^{ 2 }-6x+c=0$, then the positive value of $c$ is

  1. $2$
  2. $3$
  3. $4$
  4. $9$
  5. $8$
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E Correct answer
Explanation

For the equation x^2 - 6x + c = 0, the sum of roots is alpha + alpha^2 = 6 and the product is alpha * alpha^2 = c. Solving alpha^2 + alpha - 6 = 0 gives (alpha + 3)(alpha - 2) = 0, so alpha is 2 or -3. If alpha = 2, c = 2^3 = 8; if alpha = -3, c = (-3)^3 = -27. Since c must be positive, c = 8.

AI explanation

Using the sum of roots formula for x squared - 6x + c = 0, we have alpha plus alpha squared equals 6. By inspection, alpha equals 2 satisfies this equation because 2 plus 4 is 6. Using the product of roots formula, the positive value of c is alpha times alpha squared, which is 2 times 4, or 8.

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