Multiple choice

Let $x_1$ and $x_2$ be the roots of the equations $x^2 + px - 3 = 0$. If $x_1^2 + x_2^2 = 10$, then the value of $P$ is equal to

  1. $-4$ or $4$
  2. $-3$ or $3$
  3. $-2$ or $2$
  4. $-1$ or $4$
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C Correct answer
Explanation

For x^2 + px - 3 = 0, x1 + x2 = -p and x1 * x2 = -3. We know x1^2 + x2^2 = (x1 + x2)^2 - 2(x1 * x2). So 10 = (-p)^2 - 2(-3) = p^2 + 6. p^2 = 4, so p = 2 or -2.

AI explanation

By Vieta's formulas for x^2 + px - 3 = 0, the sum of the roots is x1 + x2 = -p and the product is x1*x2 = -3. Using the algebraic identity (x1 + x2)^2 = x1^2 + x2^2 + 2*x1*x2, we substitute the known values to get (-p)^2 = 10 + 2*(-3), which simplifies to p^2 = 4. Taking the square root gives p = 2 or p = -2.

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