Multiple choice

If α and β are the roots of the quadratic equation x2 + αx + β = 0, where β ≠ 0, then what is the value of α - β?

  1. 4

  2. 3

  3. -1

  4. -3
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For x^2 + ax + b = 0, the sum of roots is -a and product is b. So alpha + beta = -a and alpha * beta = b. Given alpha and beta are roots, alpha^2 + a*alpha + b = 0 and beta^2 + a*beta + b = 0. Substituting a = -(alpha+beta), we get alpha^2 - (alpha+beta)alpha + beta = 0 => alpha^2 - alpha^2 - alpha*beta + beta = 0 => beta(1-alpha) = 0. Since beta is not 0, alpha = 1. Then 1 + a + b = 0 and beta^2 + a*beta + b = 0. Solving these leads to alpha - beta = 3.

AI explanation

For the equation x2 + alpha x + beta = 0 with roots alpha and beta, the sum of roots gives alpha + beta = -alpha, meaning beta = -2 alpha. The product of roots gives alpha beta = beta. Since beta is not equal to zero, we divide by beta to find alpha = 1. Substituting alpha = 1 into beta = -2 alpha gives beta = -2. Therefore, the value of alpha - beta is 1 - (-2), which equals 3.

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