To solve the equation x4 + 5x2 – 14 = 0 by factoring, we can start by making a substitution to simplify the expression. Let y = x2. This transforms the equation into a quadratic form:
y2 + 5y – 14 = 0
Next, we will factor this quadratic equation. We are looking for two numbers that multiply to give -14 (the constant term) and add up to 5 (the coefficient of y). After considering the factors of -14, we find that (7, -2) works:
(y + 7)(y – 2) = 0
Now we set each factor to zero:
y + 7 = 0 or y – 2 = 0
Solving for y gives:
y = -7 or y = 2
Now we substitute back for y = x2:
For y = -7: x2 = -7 does not yield real solutions since we cannot take the square root of a negative number in the reals.
For y = 2: x2 = 2 leads to x = ±√2. Hence, the real solutions of the original equation are:
x = √2 and x = -√2.
In conclusion, the solutions for the equation x4 + 5x2 – 14 = 0 are:
x = √2 and x = -√2.