Final answer:
The equation x⁴ - 13x² + 36 = 0 can be solved using the complete the square method. The possible values of x are -3, 3, -2, and 2.
Step-by-step explanation:
This equation can be solved using the complete the square method. Let's begin:
x⁴ - 13x² + 36 = 0
First, let's rewrite the equation as a quadratic expression:
(x²)² - 13x² + 36 = 0
Now, let's complete the square by adding and subtracting the square of half the coefficient of x²:
(x²)² - 13x² + ({{-13/2}})² - ({{-13/2}})² + 36 = 0
Simplifying, we get:
(x² - 6.5)² - 42.25 + 36 = 0
Combining like terms, we have:
(x² - 6.5)² - 6.25 = 0
Now, let's solve for x:
(x² - 6.5)² = 6.25
Taking the square root of both sides, we get:
x² - 6.5 = ±√6.25
x² - 6.5 = ±2.5
Solving for x, we have two cases:
Case 1: x² - 6.5 = 2.5
x² = 9
x = ±3
Case 2: x² - 6.5 = -2.5
x² = 4
x = ±2
So, the possible values of x are x = -3, x = 3, x = -2, and x = 2.