Solve for U?

Solve for U in terms of alpha.

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  • 5 years ago
    Best Answer

    I'm going to use "a" instead of ɑ just to make things easier for me to type/copy/paste:

    Solving for "u":

    a = (1 - u) / √(1 - u²)

    start out by multiplying both sides by the denominator of that fraction:

    a√(1 - u²) = 1 - u

    Now square both sides to get the variable out of the radical:

    a²(1 - u²) = (1 - u)²

    simplify both sides, then set this up as a quadratic with "u" as your variable and "a" treated as a constant:

    a² - a²u² = 1 - 2u + u²

    -a²u² - u² + 2u + a² - 1 = 0

    I'm going to factor out a -u² from the first two terms:

    -u²(a² + 1) + 2u + a² - 1 = 0

    and finally, I'll multiply both sides by a -1 to get rid of the negative in the u² coefficient:

    u²(a² + 1) - 2u - a² + 1 = 0

    Now we can solve this using quadratic equation for u where:

    a = a² + 1

    b = -2

    c = -a² + 1

    That gives us:

    u = [ -b ± √(b² - 4ac)] / (2a)

    u = [ -(-2) ± √((-2)² - 4(a² + 1)(-a² + 1))] / (2(a² + 1))

    u = [ 2 ± √(4 - 4(-a⁴ + a² - a² + 1))] / (2a² + 2)

    u = [ 2 ± √(4 - 4(-a⁴ + 1))] / (2a² + 2)

    u = [ 2 ± √(4 + 4a⁴ - 4)] / (2a² + 2)

    u = [ 2 ± √(4a⁴)] / (2a² + 2)

    u = (2 ± 2a²) / (2a² + 2)

    Factor out a 2 from both numerator and denominator:

    u = 2(1 ± a²) / [2(a² + 1)]

    2's cancel out:

    u = (1 ± a²) / (a² + 1)

    Now let's split this into two equation, one with + and one with - to simplify:

    u = (1 - a²) / (a² + 1) and (1 + a²) / (a² + 1)

    I want to move the a² in both numerators to the front of the binomial. It's easy to do in the + version, since I can rearrange the numbers in an addition all I want. But if I do that with the - version, I inadvertently multiply the result by -1 in the process. So to fix that, I'll multiply the numerator by another -1 at the same time to make the net change a "1", which doesn't change the value.

    Now we have:

    u = -(a² - 1) / (a² + 1) and (a² + 1) / (a² + 1)

    The - side can be simplified more if you want to:

    u = (-a² + 1) / (a² + 1)

    the + side has the same numerator and denominator, so we can say "that's a 1".

    But before we declare that an answer, note what happens to the original equation if we plug a "1" in for "u".

    √(1 - u²)

    √(1 - 1²)

    √(1 - 1)

    √0

    0

    We can't have a 0 in the denominator, so u cannot be 1.

    So that leaves this as the only "solution" for this "u" in terms of alpha:

    u = (-a² + 1) / (a² + 1)

  • a = (1 - u) / sqrt(1 - u^2)

    a^2 = (1 - u)^2 / (1 - u^2)

    a^2 = (1 - u)^2 / ((1 - u) * (1 + u))

    a^2 = (1 - u) / (1 + u)

    a^2 * (1 + u) = 1 - u

    a^2 + a^2 * u = 1 - u

    a^2 * u + u = 1 - a^2

    u * (a^2 + 1) = (1 - a^2)

    u = (1 - a^2) / (1 + a^2)

  • 5 years ago

    Why can't you do it? It's stupid simple

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