To solve the equation x4 + 5x2 – 36 = 0 using factoring, we first make a substitution to simplify the equation. Let’s set y = x2. This transforms our equation into:
y2 + 5y – 36 = 0
Now, we need to factor this quadratic equation. We are looking for two numbers that multiply to -36 (the constant term) and add up to 5 (the coefficient of y). The numbers that satisfy these conditions are 9 and -4, since:
- 9 × (-4) = -36
- 9 + (-4) = 5
Thus, we can factor the quadratic as:
(y + 9)(y – 4) = 0
Setting each factor equal to zero gives us:
y + 9 = 0 or y – 4 = 0
Solving these equations yields:
- y = -9
- y = 4
We now substitute back for y = x2:
- For y = -9:
x2 = -9
This gives us no real solutions since the square of a real number cannot be negative. However, we can express the solutions in the complex number system: - x = ±3i
- For y = 4:
x2 = 4
Taking the square root of both sides, we find: - x = ±2
In summary, the solutions of the original equation x4 + 5x2 – 36 = 0 are:
- x = 2
- x = -2
- x = 3i
- x = -3i