Find all the zeroes of the equation.

-3x^4+27x^2+1200=0
Use quadratic formula and show all work.

impatient much?

Oh boy, solving quadratic equations can be a real carnival ride! Let's hop on and solve this equation step by step using the quadratic formula.

First, let's rewrite the equation in the standard quadratic form: -3x^4 + 27x^2 + 1200 = 0.

Now, let’s identify the coefficients: a = -3, b = 27, and c = 1200.

The quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a).

So, plugging in the values, we get:

x = (-(27) ± √((27)^2 - 4(-3)(1200))) / (2(-3))

Simplifying further:

x = (-27 ± √(729 + 14400)) / -6

Well, this equation looks more like a circus act than anything else! Let's continue:

x = (-27 ± √(15129)) / -6

Now, taking the square root of 15129, we get:

x = (-27 ± 123) / -6

Splitting it into two possibilities:

x₁ = (-27 + 123) / -6 = 96 / -6 = -16

x₂ = (-27 - 123) / -6 = -150 / -6 = 25

And there you have it, ladies and gentlemen! The zeroes of the equation -3x^4 + 27x^2 + 1200 = 0 are -16 and 25. Enjoy the quadratic circus!

To find the zeroes of the equation -3x^4 + 27x^2 + 1200 = 0, we can use the quadratic formula. The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are coefficients.

In our equation, a = -3, b = 27, and c = 1200. We can plug these values into the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

Let's solve step by step:

1. Identify the values of a, b, and c from the given equation.
a = -3
b = 27
c = 1200

2. Substitute the values of a, b, and c into the quadratic formula.
x = (-27 ± √(27^2 - 4*(-3)*1200)) / (2*(-3))

3. Simplify the inside of the square root.
x = (-27 ± √(729 + 14400)) / (-6)

4. Calculate the values inside the square root.
x = (-27 ± √(15129)) / (-6)

5. Take the square root of 15129.
x = (-27 ± 123) / (-6)

6. Simplify the expression.
x = (-27 + 123) / (-6) OR x = (-27 - 123) / (-6)

7. Simplify the numerators.
x = 96 / (-6) OR x = (-150) / (-6)

8. Simplify the fractions.
x = -16 OR x = 25

Therefore, the zeroes of the equation are x = -16 and x = 25.

To find the zeroes of the equation -3x^4 + 27x^2 + 1200 = 0, we can consider it as a quadratic equation in terms of x^2, and then use the quadratic formula to solve for x. Here's how you can do it step by step:

Step 1: Let's rearrange the equation as -3(x^4 - 9x^2) - 1200 = 0.

Step 2: Now, we can consider the expression inside the parentheses as a quadratic equation in terms of x^2: x^4 - 9x^2 = 0.

Step 3: Rewrite the equation: x^2(x^2 - 9) = 0.

Step 4: Set each factor equal to zero and solve them individually.

a) x^2 = 0:
Solving for x, we get x = 0.

b) x^2 - 9 = 0:
Apply the quadratic formula, where a = 1, b = 0, and c = -9.
x = (-b ± √(b^2 - 4ac)) / (2a)

Plugging in the values, we get:
x = (0 ± √(0^2 - 4(1)(-9))) / (2(1))
Simplifying further:
x = (0 ± √(0 + 36)) / 2
x = (0 ± √36) / 2
x = (0 ± 6) / 2

Simplifying the solutions, we have:
x = 6/2 = 3
x = -6/2 = -3

Step 5: Combine all the solutions:
x = 0, 3, -3

Therefore, the zeroes of the equation -3x^4 + 27x^2 + 1200 = 0 are 0, 3, and -3.