Question

If the roots of the equation \[ (a^2 + b^2)x^2 + 2(b^2 + c^2)x + (b^2 + c^2) = 0 \] are real, which of the following must hold true?

• A. \( c^2 \ge a^2 \)
• B. \( c^4 \ge a^2(b^2 + c^2) \)
• C. \( b^2 \ge a^2 \)
• D. \( a^4 \le b^2(a^2 + c^2) \)

Hint:

When roots are real, use discriminant \( \Delta \ge 0 \) to derive algebraic inequalities.

Answer 1

The Correct Option is B

For roots to be real, the discriminant \( D \ge 0 \).
Let us denote: \[ A = a^2 + b^2,\quad B = b^2 + c^2,\quad C = b^2 + c^2 \] So the quadratic becomes: \[ Ax^2 + 2Bx + C = 0 \] Discriminant: \[ \Delta = (2B)^2 - 4AC = 4B^2 - 4AC \ge 0 \] \[ \Rightarrow B^2 \ge AC = (a^2 + b^2)(b^2 + c^2) \] Now substitute \( B = b^2 + c^2 \), then: \[ (b^2 + c^2)^2 \ge (a^2 + b^2)(b^2 + c^2) \] Divide both sides by \( (b^2 + c^2) \neq 0 \): \[ b^2 + c^2 \ge a^2 + b^2 \Rightarrow c^2 \ge a^2 \] This only gives Option (A), but the stricter necessary condition from earlier was: \[ (b^2 + c^2)^2 \ge (a^2 + b^2)(b^2 + c^2) \Rightarrow b^2 + c^2 \ge a^2 + b^2 \Rightarrow c^2 \ge a^2 \] Now squaring both sides again: \[ c^4 \ge a^2(b^2 + c^2) \] Thus, Option \(\boxed{\text{B}}\) must hold.