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.