x^4-29x^2=-100 solve and show all you work and steps
a=1 b=29 c=-100
polynomial equation
To solve the equation x^4 - 29x^2 = -100, we can use substitution. Let's substitute y = x^2:
y^2 - 29y = -100
Now we have a quadratic equation. Rearranging it, we get:
y^2 - 29y + 100 = 0
We can solve this quadratic equation by factoring or by using the quadratic formula. Since the quadratic does not factor easily, we will use the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -29, and c = 100:
y = (-(-29) ± √((-29)^2 - 4(1)(100))) / (2(1))
y = (29 ± √(841 - 400)) / 2
y = (29 ± √441) / 2
y = (29 ± 21) / 2
We have two possible values for y:
y1 = (29 + 21) / 2 = 25
y2 = (29 - 21) / 2 = 4
Now we substitute back x^2 for y:
For y1 = 25:
x^2 = 25
x = ±√25
x = ±5
For y2 = 4:
x^2 = 4
x = ±√4
x = ±2
Therefore, the solutions to the equation x^4 - 29x^2 = -100 are x = -5, -2, 2, and 5.