Find all the zeroes of the equation. 27x^2−324=−x^4
To find the zeroes of the equation 27x^2 - 324 = -x^4, we first move all the terms to one side to set the equation equal to zero:
x^4 + 27x^2 - 324 = 0
Next, we can factor the equation to simplify it. Notice that this equation is in the form of a quadratic equation with x^2 in the place of the quadratic term. Let's substitute y = x^2:
y^2 + 27y - 324 = 0
Now, we can factor this quadratic equation:
(y + 36)(y - 9) = 0
Setting each factor equal to zero:
y + 36 = 0 or y - 9 = 0
Solving for y:
y = -36 or y = 9
Since we let y = x^2, we need to solve for x:
x^2 = -36 or x^2 = 9
Taking the square root of both sides:
x = ± √(-36) or x = ± √9
x = ± 6i or x = ± 3
Therefore, the zeroes of the equation are x = ± 6i and x = ± 3.