Question

2x^4-9x^3=21x^2-26x+12/2x-3

Answer 1

The = sign in your problem is an error;
it should probably be a + sign. So I
used a + sign.

Solving 4th (x^4) order Eqs is a very tedious and complex operation and is
done mostly by trial and error. By using EXCEL Program, i was able to simplify the procedure somewhat.

The process starts by dividing the Eq
by a binomial that gives a zero remainder,using long division: 2x - 3 (given) and x-1.

(2x^4 - 9x^3 + 21x^2 - 26x + 12) / (2x - 3) = x^3 - 3x^2 + 6x - 4,

(x^3 - 3x^2 + 6x - 4) / (x - 1) =
x^2 - 2x + 4,
Since we have reduced the 4th deg. Eq
to 2nd deg., we can use the Quadratic Formula to solve it.

x = (-2 +- sqrt(4 - 16)) / 2 =
(-2 +- 2i * sqrt(3)) / 2 =

x = -1 + i * sqrt(3),
x = -1 - i * sqrt(3),
How many solutions(roots) do we have?
The maximum for a 4th order(degree) Eq
is 4. Let's see:

2x - 3 = 0,
x = 3/2.

x - 1 = 0,
x = 1.

We have 4 solutions:
x = 3/2,
x = 1.
x = -1 - i * sqrt(3),
x = -1 + i * sqrt(3).
The last 2 are Imagenary.