Question

Find the nature of roots of the equation \(4x^2 - 4a^2x + a^4 - b^4 = 0\), where \(b \ne 0\).

Hint:

Use the discriminant \(D = b^2 - 4ac\) to determine the nature of roots: If \(D>0\) → real and distinct.

Answer 1

Given:
Equation: \(4x^2 - 4a^2 x + a^4 - b^4 = 0\), where \(b \neq 0\).

Step 1: Identify coefficients
\[ A = 4, \quad B = -4a^2, \quad C = a^4 - b^4 \]

Step 2: Calculate discriminant \(D\)
\[ D = B^2 - 4AC = (-4a^2)^2 - 4 \times 4 \times (a^4 - b^4) = 16a^4 - 16(a^4 - b^4) \] \[ = 16a^4 - 16a^4 + 16b^4 = 16b^4 \]

Step 3: Analyze discriminant
Since \(b \neq 0\), \(b^4 > 0\), so
\[ D = 16b^4 > 0 \]

Step 4: Conclusion
Because \(D > 0\), the equation has two distinct real roots.

Final Answer:
The roots are real and distinct.