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 try to factor it. However, it is not an obvious factoring problem.
Let's make a substitution to ease the calculations.
Let's set y = x^2.
Now the equation becomes y^2 - 29y = 100.
Rearranging the terms, we have y^2 - 29y - 100 = 0.
Now we have a quadratic equation. Let's solve it using the quadratic formula.
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions are given by
x = (-b ± √(b^2 - 4ac))/(2a)
In our case, a = 1, b = -29, and c = -100.
Plugging these values into the quadratic formula, we get:
y = (-(-29) ± √((-29)^2 - 4(1)(-100))) / (2*1)
y = (29 ± √(841 + 400)) / 2
y = (29 ± √(1241)) / 2
Now, we need to simplify the square root term. We know that 1241 is not a perfect square, so we can't simplify it further.
Now we have two possibilities:
y = (29 + √(1241)) / 2 or
y = (29 - √(1241)) / 2
Now, recall that y = x^2.
So, for the first possibility, we have:
(29 + √(1241)) / 2 = x^2
To isolate x, we can take the square root of both sides:
sqrt((29 + √(1241)) / 2) = x
Now, for the second possibility, we have:
(29 - √(1241)) / 2 = x^2
Taking the square root of both sides:
sqrt((29 - √(1241)) / 2) = x
So the two solutions to the equation x^4 - 29x^2 = 100 are:
x = sqrt((29 + √(1241)) / 2) or x = sqrt((29 - √(1241)) / 2)