Multiple choice

If the equations $x^2 + ax + 1 =0$ and $x^2 - x - a = 0$ have a real common root b, then the value of b is equal to

  1. $0$
  2. $1$
  3. $-1$
  4. $2$
  5. $3$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Subtract the two equations: (x^2 + ax + 1) - (x^2 - x - a) = 0 => x(a+1) + (1+a) = 0 => (x+1)(a+1) = 0. If a = -1, the equations are identical (x^2 - x + 1 = 0), which has no real roots. Thus, x+1 = 0, so x = -1. If x = -1, (-1)^2 + a(-1) + 1 = 0 => 2 - a = 0 => a = 2.

AI explanation

If the two equations x^2 + ax + 1 = 0 and x^2 - x - a = 0 share a common real root b, we can set them equal to each other for that root. This gives b^2 + ab + 1 = b^2 - b - a. Subtracting b^2 from both sides gives ab + 1 = -b - a. Rearranging terms gives ab + a + b + 1 = 0. Factoring by grouping yields a(b+1) + (b+1) = 0, or (a+1)(b+1) = 0. Assuming a is not equal to -1, we must have b = -1.

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