Multiple choice

Find a if equations $x^3+ax+1=0$ and $x^4+ax^2+1=0$ have a common root

  1. 0

  2. 1

  3. -2

  4. 2
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

If x is a common root, x^3 + ax + 1 = 0 and x^4 + ax^2 + 1 = 0. Multiplying the first by x gives x^4 + ax^2 + x = 0. Subtracting this from the second equation gives 1 - x = 0, so x = 1. Substituting x = 1 into the first equation: 1 + a + 1 = 0, so a = -2.

AI explanation

If the equations share a common root, we can express it as r^3 = -ar - 1 and r^4 = -ar^2 - 1. Multiplying the first equation by r yields r^4 = -ar^2 - r, and equating the two expressions for r^4 gives r = 1. Substituting r = 1 back into the first equation results in 1 = -a - 1, which solves to a = -2. This value of a is the required solution.

Keep practicing — related questions