Question

If the equation \( x^4 + ax^3 + bx^2 + cx + d = 0 \) has three equal roots, then the root is:

• A. \( \frac{6c - ab}{8b - 3a^2} \)
• B. \( \frac{ab - 6c}{8b + 3a^2} \)
• C. \( \frac{6c - ab}{3a^2 - 4b} \)
• D. \( \frac{6c - ab}{3a^2 - 8b} \)

Hint:

Match the coefficients to solve for the root in equations with repeated roots.

Answer 1

The Correct Option is A

For three equal roots, factor as \( (x - r)^3(x - s) = 0 \). Expanding: \[ x^4 - (3r + s)x^3 + (3r^2 + 2rs)x^2 - (r^3 + 3r^2s)x + r^3s = 0 \] Matching coefficients with \( x^4 + ax^3 + bx^2 + cx + d = 0 \): \[ a = -(3r + s), \quad b = 3r^2 + 2rs, \quad c = -(r^3 + 3r^2s), \quad d = r^3s \] From \( a = -(3r + s) \), solve for \( s = -a - 3r \). Substituting into \( b \) and \( c \), we get: \[ r = \frac{6c - ab}{8b - 3a^2} \] Thus, the root is \( \frac{6c - ab}{8b - 3a^2} \), option (1).