Multiple choice

If the equations $x^2 \, +\, ax \, +\,b\, =\, 0$ and $x^2 \, +\, bx \, +\, a\, =\, 0$ have a common root, then the value of $a + b$ is:

  1. $+1$
  2. $0$
  3. $-1$
  4. $2$
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C Correct answer
Explanation

If the equations have a common root alpha, then alpha^2 + a*alpha + b = 0 and alpha^2 + b*alpha + a = 0. Subtracting the two gives (a-b)*alpha + (b-a) = 0, so (a-b)(alpha - 1) = 0. If a != b, then alpha = 1. Substituting alpha = 1 into the first equation gives 1 + a + b = 0, so a + b = -1.

AI explanation

Let alpha be the common root of the equations x^2 + ax + b = 0 and x^2 + bx + a = 0. Subtracting the second equation from the first gives (a - b)alpha + (b - a) = 0, which factors to (a - b)(alpha - 1) = 0. Assuming a and b are distinct, the common root must be 1. Substituting x = 1 into either equation gives 1 + a + b = 0, meaning a + b = -1.

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