Question:
If \( x_1 \) and \( x_2 \) are the roots of \( 3x^2 - 2x - 6 = 0 \), then \( x_1^2 + x_2^2 \) is equal to:
If \( x_1 \) and \( x_2 \) are the roots of \( 3x^2 - 2x - 6 = 0 \), then \( x_1^2 + x_2^2 \) is equal to:
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The identity \( a^2 + b^2 = (a+b)^2 - 2ab \) is essential for symmetric root problems. Always find the sum and product first before attempting to evaluate the specific expression.
Updated On: May 1, 2026
- \( \frac{50}{9} \)
- \( \frac{40}{9} \)
- \( \frac{30}{9} \)
- \( \frac{20}{9} \)
- \( \frac{10}{9} \)
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The Correct Option is B
Solution and Explanation
Concept: For a quadratic equation \( ax^2 + bx + c = 0 \), the sum of roots (\( S \)) and product of roots (\( P \)) are given by: \[ S = x_1 + x_2 = -\frac{b}{a}, \quad P = x_1 x_2 = \frac{c}{a} \] The expression \( x_1^2 + x_2^2 \) can be rewritten using these identities as \( (x_1 + x_2)^2 - 2x_1x_2 \).
Step 1: Finding the sum and product of the roots.
From the equation \( 3x^2 - 2x - 6 = 0 \), we identify \( a=3, b=-2, c=-6 \): \[ x_1 + x_2 = -\frac{-2}{3} = \frac{2}{3} \] \[ x_1 x_2 = \frac{-6}{3} = -2 \]
Step 2: Calculating \( x_1^2 + x_2^2 \).
Substitute the values into the identity: \[ x_1^2 + x_2^2 = \left( \frac{2}{3} \right)^2 - 2(-2) \] \[ = \frac{4}{9} + 4 = \frac{4 + 36}{9} = \frac{40}{9} \]
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