To find the zeros of the equation -3x^4 + 27x^2 + 1200 = 0, we can use factoring and the quadratic formula. Here's how you can do it step by step:
Step 1: Let's start by factoring out a common factor if possible. In this case, we can factor out -3 from all terms:
-3(x^4 - 9x^2 - 400) = 0
Step 2: Now, let's focus on the expression in the parentheses: x^4 - 9x^2 - 400. To simplify this expression, we can substitute x^2 with a new variable, let's say y:
y^2 - 9y - 400 = 0
Step 3: Solve the quadratic equation y^2 - 9y - 400 = 0 by factoring or applying the quadratic formula. In this case, factoring doesn't work, so we'll use the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 1, b = -9, and c = -400. Plugging these values into the quadratic formula, we get:
y = [9 ± √((-9)^2 - 4(1)(-400))] / (2*1)
= [9 ± √(81 + 1600)] / 2
= (9 ± √(1681)) / 2
= (9 ± 41) / 2
Therefore, we have two possible values for y:
y1 = (9 + 41) / 2 = 50/2 = 25
y2 = (9 - 41) / 2 = -32/2 = -16
Step 4: Now, we have found the values of y. We need to substitute them back into the equation y = x^2:
For y1 = 25: x^2 = 25
Taking the square root of both sides, we get:
x1 = √25 = ±5
For y2 = -16: x^2 = -16
Since we can't take the square root of a negative number in real numbers, this value does not yield real solutions.
Therefore, the zeros or solutions to the original equation -3x^4 + 27x^2 + 1200 = 0 are x = ±5.
Please note that complex solutions may exist in other cases, but in this particular question, we don't have any.