Can someone help me find my solution(s) to this cubic equation?
x = a(1-t)^3 + 3bt(1-t)^2 + 3c(1-t)t^2 + dt^3
Where:
a = 1
b = 3
c = 8
d = 9
x = 5
Ultimately, I need to have the solutions with the variables in place so I can easily figure out how to substitute other constants in.
I've been working on this for weeks and I'm struggling! I've looked up cubic functions on wikipedia and I've gone back and reread both an algebra I and an algebra II books and I'm just not able to apply what I've found to this!
Basically, I need to be able to solve for t when I have x so I need an equation where t = MyCalculation. I understand that there could possibly be up to 3 solutions but in my circumstance there should never be more than 1 since my curve should never cross the same x or y plane more than once.
If you're looking for a nice solution in terms of $a,b,c,d,$ and $x$, you're going to be disappointed. I used Maple to find solutions. For simplicity I'm only going to list the solutions that have no explicitly stated imaginary component.
If you restrict $a,b,c,$ and $d$ to be the values you gave, then
$t=1/14\,\sqrt [3]{916-196\,x+28\,\sqrt {77-458\,x+49\,{x}^{2}}}+{\frac { 46}{7}}\,{\frac {1}{\sqrt [3]{916-196\,x+28\,\sqrt {77-458\,x+49\,{x}^ {2}}}}}+3/7$
Otherwise, you get this hideously messy solution (sorry if the formatting is horrible. There's really no way around that.)
$1/2\,{\frac {\sqrt [3]{24\,cdx+24\,a{c}^{2}-36\,x{c}^{2}+4\,\sqrt {6\, {x}^{2}bd+4\,a{c}^{3}-3\,{b}^{2}{c}^{2}+9\,{x}^{2}{c}^{2}-6\,acbd+6\,a xcb-18\,{x}^{2}bc+6\,a{x}^{2}c-12\,ax{c}^{2}-2\,{a}^{2}xd+{a}^{2}{d}^{ 2}+4\,{b}^{3}d+{x}^{2}{d}^{2}-4\,{b}^{3}x-4\,{c}^{3}x+{a}^{2}{x}^{2}+6 \,cbdx+6\,acdx+6\,abdx-12\,{b}^{2}dx+6\,xc{b}^{2}+6\,xb{c}^{2}-2\,a{x} ^{2}d-2\,ax{d}^{2}-6\,cd{x}^{2}+9\,{x}^{2}{b}^{2}-6\,a{x}^{2}b}a-12\, \sqrt {6\,{x}^{2}bd+4\,a{c}^{3}-3\,{b}^{2}{c}^{2}+9\,{x}^{2}{c}^{2}-6 \,acbd+6\,axcb-18\,{x}^{2}bc+6\,a{x}^{2}c-12\,ax{c}^{2}-2\,{a}^{2}xd+{ a}^{2}{d}^{2}+4\,{b}^{3}d+{x}^{2}{d}^{2}-4\,{b}^{3}x-4\,{c}^{3}x+{a}^{ 2}{x}^{2}+6\,cbdx+6\,acdx+6\,abdx-12\,{b}^{2}dx+6\,xc{b}^{2}+6\,xb{c}^ {2}-2\,a{x}^{2}d-2\,ax{d}^{2}-6\,cd{x}^{2}+9\,{x}^{2}{b}^{2}-6\,a{x}^{ 2}b}b+12\,\sqrt {6\,{x}^{2}bd+4\,a{c}^{3}-3\,{b}^{2}{c}^{2}+9\,{x}^{2} {c}^{2}-6\,acbd+6\,axcb-18\,{x}^{2}bc+6\,a{x}^{2}c-12\,ax{c}^{2}-2\,{a }^{2}xd+{a}^{2}{d}^{2}+4\,{b}^{3}d+{x}^{2}{d}^{2}-4\,{b}^{3}x-4\,{c}^{ 3}x+{a}^{2}{x}^{2}+6\,cbdx+6\,acdx+6\,abdx-12\,{b}^{2}dx+6\,xc{b}^{2}+ 6\,xb{c}^{2}-2\,a{x}^{2}d-2\,ax{d}^{2}-6\,cd{x}^{2}+9\,{x}^{2}{b}^{2}- 6\,a{x}^{2}b}c-4\,\sqrt {6\,{x}^{2}bd+4\,a{c}^{3}-3\,{b}^{2}{c}^{2}+9 \,{x}^{2}{c}^{2}-6\,acbd+6\,axcb-18\,{x}^{2}bc+6\,a{x}^{2}c-12\,ax{c}^ {2}-2\,{a}^{2}xd+{a}^{2}{d}^{2}+4\,{b}^{3}d+{x}^{2}{d}^{2}-4\,{b}^{3}x -4\,{c}^{3}x+{a}^{2}{x}^{2}+6\,cbdx+6\,acdx+6\,abdx-12\,{b}^{2}dx+6\,x c{b}^{2}+6\,xb{c}^{2}-2\,a{x}^{2}d-2\,ax{d}^{2}-6\,cd{x}^{2}+9\,{x}^{2 }{b}^{2}-6\,a{x}^{2}b}d+24\,xab-24\,xbd-12\,bad-12\,cad-12\,cbd+8\,xad +8\,{b}^{3}+8\,{c}^{3}+4\,{a}^{2}d+24\,{b}^{2}d-4\,x{a}^{2}-4\,x{d}^{2 }+4\,a{d}^{2}-12\,c{b}^{2}-12\,b{c}^{2}-12\,abc-24\,xac+72\,xcb-36\,x{ b}^{2}}}{a-3\,b+3\,c-d}}-2\,{\frac {ac-ad-{b}^{2}+cb+bd-{c}^{2}}{ \left( a-3\,b+3\,c-d \right) \sqrt [3]{24\,cdx+24\,a{c}^{2}-36\,x{c}^ {2}+4\,\sqrt {6\,{x}^{2}bd+4\,a{c}^{3}-3\,{b}^{2}{c}^{2}+9\,{x}^{2}{c} ^{2}-6\,acbd+6\,axcb-18\,{x}^{2}bc+6\,a{x}^{2}c-12\,ax{c}^{2}-2\,{a}^{ 2}xd+{a}^{2}{d}^{2}+4\,{b}^{3}d+{x}^{2}{d}^{2}-4\,{b}^{3}x-4\,{c}^{3}x +{a}^{2}{x}^{2}+6\,cbdx+6\,acdx+6\,abdx-12\,{b}^{2}dx+6\,xc{b}^{2}+6\, xb{c}^{2}-2\,a{x}^{2}d-2\,ax{d}^{2}-6\,cd{x}^{2}+9\,{x}^{2}{b}^{2}-6\, a{x}^{2}b}a-12\,\sqrt {6\,{x}^{2}bd+4\,a{c}^{3}-3\,{b}^{2}{c}^{2}+9\,{ x}^{2}{c}^{2}-6\,acbd+6\,axcb-18\,{x}^{2}bc+6\,a{x}^{2}c-12\,ax{c}^{2} -2\,{a}^{2}xd+{a}^{2}{d}^{2}+4\,{b}^{3}d+{x}^{2}{d}^{2}-4\,{b}^{3}x-4 \,{c}^{3}x+{a}^{2}{x}^{2}+6\,cbdx+6\,acdx+6\,abdx-12\,{b}^{2}dx+6\,xc{ b}^{2}+6\,xb{c}^{2}-2\,a{x}^{2}d-2\,ax{d}^{2}-6\,cd{x}^{2}+9\,{x}^{2}{ b}^{2}-6\,a{x}^{2}b}b+12\,\sqrt {6\,{x}^{2}bd+4\,a{c}^{3}-3\,{b}^{2}{c }^{2}+9\,{x}^{2}{c}^{2}-6\,acbd+6\,axcb-18\,{x}^{2}bc+6\,a{x}^{2}c-12 \,ax{c}^{2}-2\,{a}^{2}xd+{a}^{2}{d}^{2}+4\,{b}^{3}d+{x}^{2}{d}^{2}-4\, {b}^{3}x-4\,{c}^{3}x+{a}^{2}{x}^{2}+6\,cbdx+6\,acdx+6\,abdx-12\,{b}^{2 }dx+6\,xc{b}^{2}+6\,xb{c}^{2}-2\,a{x}^{2}d-2\,ax{d}^{2}-6\,cd{x}^{2}+9 \,{x}^{2}{b}^{2}-6\,a{x}^{2}b}c-4\,\sqrt {6\,{x}^{2}bd+4\,a{c}^{3}-3\, {b}^{2}{c}^{2}+9\,{x}^{2}{c}^{2}-6\,acbd+6\,axcb-18\,{x}^{2}bc+6\,a{x} ^{2}c-12\,ax{c}^{2}-2\,{a}^{2}xd+{a}^{2}{d}^{2}+4\,{b}^{3}d+{x}^{2}{d} ^{2}-4\,{b}^{3}x-4\,{c}^{3}x+{a}^{2}{x}^{2}+6\,cbdx+6\,acdx+6\,abdx-12 \,{b}^{2}dx+6\,xc{b}^{2}+6\,xb{c}^{2}-2\,a{x}^{2}d-2\,ax{d}^{2}-6\,cd{ x}^{2}+9\,{x}^{2}{b}^{2}-6\,a{x}^{2}b}d+24\,xab-24\,xbd-12\,bad-12\,ca d-12\,cbd+8\,xad+8\,{b}^{3}+8\,{c}^{3}+4\,{a}^{2}d+24\,{b}^{2}d-4\,x{a }^{2}-4\,x{d}^{2}+4\,a{d}^{2}-12\,c{b}^{2}-12\,b{c}^{2}-12\,abc-24\,xa c+72\,xcb-36\,x{b}^{2}}}}+{\frac {a-2\,b+c}{a-3\,b+3\,c-d}} $
If you're ok with an approximate solution, then I suggest you try some computational methods rather than using the exact solution.