Question:

Math, and some hard stuff too =\?

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I'm having a hard time figuring out a problem

y=ax+b√(xc+d√(f²-x²)+e)

I need to either isolate the x variable or even just turning the entire thing into a quartic equation.

My overall goal though is making a series of equations that will tell you what x needs to equal for y to equal the minimum possible

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  1. If you're trying to minimize y, solving for x won't help.  You need to take the derivative, find its roots, and select the root that gives the minimal y value.

    Edit:

    The derivative is

    dy/dx = a + bh' / 2√h

    = a + b(c + dg' / 2√g) / 2√(cx + d√g + e)

    = a + b(c - dx / √g) / 2√(cx + d√g + e)

    where g = f² - x²

    Now,

    dy/dx = 0

    -a = b(c - dx / √g) / 2√(cx + d√g + e)

    a² = b²(c - dx / √g)² / 4(cx + d√g + e)

    4a²(cx + d√g + e) / b² = c² - 2cdx / √g + d²x² / g

    4a²cx + 4a²d√g + 4a²e = b²c² - 2b²cdx / √g + b²d²x² / g

    4a²cgx + 4a²dg√g + 4a²eg = b²c²g - 2b²cdx√g + b²d²x²

    4a²cgx  + 4a²eg - b²c²g - b²d²x² = -2b²cdx√g - 4a²dg√g

    (4a²cgx  + 4a²eg - b²c²g - b²d²x²)² = g(2b²cdx + 4a²dg)²

    The implicit derivative might have been cubic in x and y, but this solution looks to be hexic.  So it looks to me like your only hope is a numeric approach.  Hope that doesn't spoil your day.

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