I do not know English well, but the shape will help you fill in any gaps.
Let's put
\(\displaystyle c = -70.25, d = 5, e = 240\)
\(\displaystyle f(x) = x ^ 4 + cx ^ 2 + dx + e = 0\)
There are known criteria that ensure that \(\displaystyle f\) has two positive and two negative roots (
https://en.m.wikipedia.org/wiki/Quartic_function - "Nature of the roots"). The interest in this case is that the equation accepts a simple geometric interpretation from which the below method of resolution arises. This means that we will first find the values of the basic parameters of this model, and then we will get the analytical solutions using geometric terms. See the image (the center of the red cross is near the beginning of the axes and not exactly there, ie the shape is right).
One corollary of this interpretation is that the roots of \(\displaystyle f\) will lie within the limits
\(\displaystyle -R \leq x \leq R\)
where
\(\displaystyle R = \sqrt {\frac {d ^ 2} {4e} - c} \approx 8.38308\)
As a resolvent of \(\displaystyle f\) we will use the equation
\(\displaystyle g(z) = z ^ 3 - cz ^ 2 - 4ez + 4ec - d ^ 2 = 0\)
with solutions
\(\displaystyle z_1 \approx -70.24371, z_2 \approx -30.99414, z_3 \approx 30.98785\)
We will then calculate the angles
\(\displaystyle \omega_n = \cos ^ {-1} \sqrt {\frac {1} {2} + \sqrt {\frac {1} {4} - \frac {e} {z_n ^ 2}}}\)
for \(\displaystyle n = 1, 2, 3\).
so we will receive
\(\displaystyle \omega_1 \approx 13.08676 ^ \circ, \omega_2 \approx 44.26230 ^ \circ, \omega_3 \approx 44.54057 ^ \circ\)
For the purpose of designing the model of \(\displaystyle f\), note that it must be \(\displaystyle \omega_n \leq 45 ^ \circ\). If the angle forming a colored system with the \(\displaystyle y-\)axis is equal with \(\displaystyle \omega\), then the \(\displaystyle 90 ^ \circ - \omega\) angle will also be formed. The above relationships calculate the value of the smallest of these angles.
Now the solutions of \(\displaystyle f\) are given by the relations:
\(\displaystyle x_1 = +R \sin (- \omega_1 - \omega_2 - \omega_3) \approx -8.20323\)
\(\displaystyle x_2 = -R \sin (+ \omega_1 - \omega_2 - \omega_3) \approx 8.12392\)
\(\displaystyle x_3 = -R \sin (- \omega_1 + \omega_2 - \omega_3) \approx 1.93778\)
\(\displaystyle x_4 = +R \sin (- \omega_1 - \omega_2 + \omega_3) \approx -1.85847\)
If \(\displaystyle d\) was negative then the roots of \(\displaystyle f\) had to be reversed as the resolvent \(\displaystyle g = 0\) does not know the sign of \(\displaystyle d\).
To design the model of \(\displaystyle f\), we still need the values of the following parameters:
\(\displaystyle a_n = \sqrt {z_n + R ^ 2}\)
for \(\displaystyle n = 1, 2, 3\).
So we will receive
\(\displaystyle a_1 \approx 0.17981, a_2 \approx 6.26753, a_3 \approx 10.06300\)
You can make your own examples and see how the model responds. It always works.
