Determine zeros of F(x)= -x^4 + x^2 + 1
x^4 -x^2 -1 = 0
let z = x^2
z^2 - z - 1 = 0
z = [ 1 +/- sqrt (1+4) ]/2
z = .5 +/- .5 sqrt 5
that means
x^2 = .5 +/- .5 sqrt 5 = 1.62 or -.618
if x^2 = 1.62
x = +/- 1.27
if x^2 = -.618 = -1 * .618
x = i * .786 or -i * .786
x =
To determine the zeros of the function F(x) = -x^4 + x^2 + 1, we need to find the values of x for which F(x) equals zero.
So, let's set up the equation:
-x^4 + x^2 + 1 = 0
One approach to solve this equation is by factoring. However, this equation is a quadratic in terms of x^2, so we can use a substitution to simplify it.
Let's substitute y = x^2:
-y^2 + y + 1 = 0
Now, we can solve this quadratic equation. There are different methods to solve quadratic equations, such as factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / 2a
For our equation, a = -1, b = 1, and c = 1. Let's substitute these values into the quadratic formula:
y = (-1 ± √(1 - 4(-1)(1))) / (2(-1))
= (-1 ± √(1 + 4)) / (-2)
= (-1 ± √5) / (-2)
Now that we have the value of y, we can substitute it back into our substitution equation to find the values of x:
y = x^2
Substituting (-1 + √5) / (-2) for y:
(-1 + √5) / (-2) = x^2
x^2 = (-1 + √5) / (-2)
And substituting (-1 - √5) / (-2) for y:
(-1 - √5) / (-2) = x^2
x^2 = (-1 - √5) / (-2)
To get the values of x, we can take the square root of both sides:
x = ±√((-1 + √5) / (-2))
x = ±√((-1 - √5) / (-2))
These are the zeros of the function F(x).