I graphed y=4x^3-70x^2+304x-175. Perhaps that would help. I think there are a couple of solutions.
Is there a mathematical way to solve this without using the graphing calculator?
A graphical solution is already a mathematical solution. In fact, the graphical solution gives three real roots, namely near x=1, x=6 and x=11.
On the basis that 2*11>16, 11 cannot be retained as a valid solution.
So the remaining real solutions are around 1 and 6.
If you are looking for an analytical way to solve the cubic, there is the Nicolo Fontana Tartaglia method which is described in detail in:
The solution consists of three steps:
1. Find the depressed equation
by removing the x² term of the general cubic by substitution, i.e.
substitute x=y-b/(3a) to give
f(y)=a*y^3 + (c-(b^2)/(3*a))*y + d-(b*c)/(3*a)+(2*b^3)/(27*a^2)
Note the absence of the y² term.
We will denote the depressed equation as
2. Find s and t such that
then a real solution of the cubic is
y=s-t from which x can be found.
3. Using long division, reduce the cubic to a quadratic and find the two remaining solutions.
Give it a try and the three (real) roots should be around 1, 6 and 11.
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