To solve the equation x4 + 5x2 – 36 = 0 using factoring, we start by making a substitution to simplify our approach. Let’s set y = x2. This transforms our equation into a quadratic form:
y2 + 5y – 36 = 0
Next, we will factor the quadratic equation. We need to find two numbers that multiply to -36 (the constant term) and add up to 5 (the coefficient of y). The numbers that meet these criteria are 9 and -4.
Thus, we can factor the quadratic as follows:
(y + 9)(y – 4) = 0
Now, we will set each factor equal to zero to find the solutions for y:
- y + 9 = 0 => y = -9
- y – 4 = 0 => y = 4
Now, we can substitute back for y:
- From y = -9:
x2 = -9
Since this results in a negative number, there are no real solutions; instead, we get complex solutions: x = 3i and x = -3i. - From y = 4:
x2 = 4
thus, x = 2 and x = -2.
In summary, the complete set of solutions for the original equation x4 + 5x2 – 36 = 0 is:
- x = 2
- x = -2
- x = 3i
- x = -3i
Therefore, the solutions include both real and complex numbers.