9x^4-24x^3+16x^2 (Find all solutions!)
x = 0 is one of them. Factoring out x^2 leaves you with the binomial equation
9x^2 -24x +16 = 0, which can be factored to
(3x -4)^2 = 0
That tells you that the other root is x = 4/3.
Normally, a 4th order equation of this type would have four solutions, some of which may be complex. This one has two real "double roots".
By the way, this is not trigonometry. It is algebra.
To find the solutions of the equation 9x^4 - 24x^3 + 16x^2 = 0, we first notice that each term has a common factor of x^2. Factoring out x^2, we get:
x^2(9x^2 - 24x + 16) = 0
Now, we can solve this equation by setting each factor equal to zero.
1) x^2 = 0
When x^2 = 0, we have x = 0 as a solution.
2) 9x^2 - 24x + 16 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, let's use 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)
For our equation 9x^2 - 24x + 16 = 0, a = 9, b = -24, and c = 16. Plugging these values into the quadratic formula, we get:
x = (-(-24) ± √((-24)^2 - 4(9)(16))) / (2(9))
Simplifying further:
x = (24 ± √(576 - 576)) / 18
x = (24 ± √0) / 18
x = (24 ± 0) / 18
Since the discriminant is zero (√0 = 0), the solutions are equal.
x = 24 / 18
x = 4 / 3
Therefore, the solutions to the equation 9x^4 - 24x^3 + 16x^2 = 0 are x = 0 and x = 4 / 3.