Answer
Option D is correct.
x = ±i√(5) OR ±i√(3)
Step-by-step explanation
The question wants us to solve
x⁴ + 8x² + 15 = 0
To solve this, we first say that
Let x² = y
So that,
x⁴ = (x²)² = y²
So, the equation becomes
y² + 8y + 15 = 0
This is a simple quadratic equation, we then solve this
y² + 8y + 15 = 0
y² + 3y + 5y + 15 = 0
y (y + 3) + 5 (y + 3) = 0
(y + 5) (y + 3) = 0
y + 5 = 0 OR y + 3 = 0
y = -5 OR y = -3
But, Recall that x² = y
If y = -5
x² = y = -5
x² = -5
x = √(-5)
If y = -3
x² = y = -3
x² = -3
x = √(-3)
So,
x = √(-5) OR x = √(-3)
Note that
√(-1) = i
√(-5) = √(-1) × √(5)
= i√5
And
√(-3) = √(-1) × √(3)
= i√3
Hence
x = ±i√(5) OR ±i√(3)
Hope this Helps!!!