Question

If the equation \[ x^4-ax^2+9=0 \] has four real and distinct roots, then the least possible integral value of $a$ is

Hint:

For biquadratic equations, always convert to quadratic in $x^2$ and check positivity of roots.

Answer 1

Correct Answer: 7

Step 1: Given equation

x⁴ − ax² + 9 = 0

We are required to find the least integer value of a such that the equation has four real and distinct roots.

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Step 2: Reduce the equation

Let

y = x²,   y ≥ 0

Then the equation becomes:

y² − ay + 9 = 0

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Step 3: Conditions for four real and distinct roots in x

For x to have four real and distinct roots, the quadratic in y must:

• Have two distinct real roots
• Both roots must be positive

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Step 4: Apply discriminant condition

Discriminant:

Δ = a² − 36

For two distinct real roots:

a² − 36 > 0

|a| > 6

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Step 5: Apply positivity condition

Product of roots:

y₁y₂ = 9 > 0

Sum of roots:

y₁ + y₂ = a

For both roots to be positive:

a > 0

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Step 6: Combine conditions

a > 6

Hence, the least integer value satisfying the condition is:

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Final Answer:

a = 7