Question

Let \( x_1 \) and \( x_2 \) be the roots of the equation \( x^2 + px - 3 = 0 \). If \( x_1^2 + x_2^2 = 10 \), then the value of \( p \) is equal to:

• A. \( -4 \) or \( 4 \)
• B. \( -3 \) or \( 3 \)
• C. \( -2 \) or \( 2 \)
• D. \( -1 \) or \( 1 \)
• E. \( 0 \)

Hint:

Squaring a variable like \( (-p)^2 \) always results in a positive value. When solving \( p^2 = k \), remember to include both the positive and negative square roots.

Answer 1

The Correct Option is C

Concept: This problem reverses the logic of the previous one. We use the given value of the sum of squares of the roots to solve for an unknown coefficient in the original quadratic equation using Vieta's formulas.

Step 1: Expressing the roots in terms of \( p \).
From \( x^2 + px - 3 = 0 \), we have: \[ x_1 + x_2 = -p, \quad x_1 x_2 = -3 \]

Step 2: Using the given condition to solve for \( p \).
We are given \( x_1^2 + x_2^2 = 10 \). Applying the identity: \[ (x_1 + x_2)^2 - 2x_1x_2 = 10 \] \[ (-p)^2 - 2(-3) = 10 \] \[ p^2 + 6 = 10 \implies p^2 = 4 \implies p = \pm 2 \]