Multiple choice

If $\alpha$ & $\beta$ are roots of the equation $x^{2}+6x+\lambda=0$ & $3\alpha+2\beta=-20$ find $\lambda$..

  1. $16$
  2. $-8$
  3. $-16$
  4. $8$
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C Correct answer
Explanation

Roots alpha, beta satisfy alpha + beta = -6 and alpha * beta = lambda. Given 3alpha + 2beta = -20. From alpha + beta = -6, beta = -6 - alpha. Substitute: 3alpha + 2(-6 - alpha) = -20 => 3alpha - 12 - 2alpha = -20 => alpha = -8. Then beta = -6 - (-8) = 2. lambda = alpha * beta = -8 * 2 = -16.

AI explanation

Using Vieta's formulas for the equation x^2 + 6x + lambda = 0, the sum of the roots alpha + beta equals -6. We also have the given linear equation 3(alpha) + 2(beta) = -20. Multiplying the sum of the roots by 2 gives 2(alpha) + 2(beta) = -12. Subtracting this result from the given linear equation yields alpha = -8. Substituting alpha back into the sum gives beta = 2. Finally, the product of the roots alpha times beta equals lambda, so lambda equals -8 times 2, which is -16.

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