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Math Help - Solving equations with rational terms.

  1. #1
    ADY
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    Solving equations with rational terms.

    Bit of help please on this fine Sunday morning

    \frac{x}{3}=\frac{2}{x}

    \frac{2}{x + 1}=\frac{2x}{2x - 3} + 3

    Rationals never where my strong point

    Thank you!
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  2. #2
    Flow Master
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    Quote Originally Posted by ADY View Post
    Bit of help please on this fine Sunday morning


    \frac{x}{3}=\frac{2}{x} Mr F says: Cross multiply: {\color{red}x^2 = 6}. Solve this.

    \frac{2}{x + 1}=\frac{2x}{2x - 3} + 3 Mr F says: Multiply both sides by (x + 1)(2x - 3):

    {\color{red}2(2x - 3) = 2x(x+1) + 3(x + 1)(2x - 3)}. Expand. Simplify. Solve.

    Rationals never where my strong point

    Thank you!
    ..
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  3. #3
    ADY
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    x = 2.449489743 x = 2

    and

    x=1

    ?
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  4. #4
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    Quote Originally Posted by ADY View Post
    x = 2.449489743 x = 2

    and

    x=1

    ?
    How are you getting three solutions to a quadratic equation?

    Please reply showing your work, so we can see where you're going astray. Thank you!
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  5. #5
    ADY
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    Simplifying
    x2 = 6

    Solving
    x2 = 6

    Move all terms containing x to the left, all other terms to the right.

    Simplifying
    x2 = 6

    Take the square root of each side:

    x = {2.449489743}
    __________________________________________________ _________________

    Simplifying
    2(2x + -3) = 2x(x + 1) + 3(x + 1)(2x + -3)

    Reorder the terms:
    2(-3 + 2x) = 2x(x + 1) + 3(x + 1)(2x + -3)
    (-3 * 2 + 2x * 2) = 2x(x + 1) + 3(x + 1)(2x + -3)
    (-6 + 4x) = 2x(x + 1) + 3(x + 1)(2x + -3)

    Reorder the terms:
    -6 + 4x = 2x(1 + x) + 3(x + 1)(2x + -3)
    -6 + 4x = (1 * 2x + x * 2x) + 3(x + 1)(2x + -3)
    -6 + 4x = (2x + 2x2) + 3(x + 1)(2x + -3)

    Reorder the terms:
    -6 + 4x = 2x + 2x2 + 3(1 + x)(2x + -3)

    Reorder the terms:
    -6 + 4x = 2x + 2x2 + 3(1 + x)(-3 + 2x)

    Multiply (1 + x) * (-3 + 2x)
    -6 + 4x = 2x + 2x2 + 3(1(-3 + 2x) + x(-3 + 2x))
    -6 + 4x = 2x + 2x2 + 3((-3 * 1 + 2x * 1) + x(-3 + 2x))
    -6 + 4x = 2x + 2x2 + 3((-3 + 2x) + x(-3 + 2x))
    -6 + 4x = 2x + 2x2 + 3(-3 + 2x + (-3 * x + 2x * x))
    -6 + 4x = 2x + 2x2 + 3(-3 + 2x + (-3x + 2x2))

    Combine like terms: 2x + -3x = -1x
    -6 + 4x = 2x + 2x2 + 3(-3 + -1x + 2x2)
    -6 + 4x = 2x + 2x2 + (-3 * 3 + -1x * 3 + 2x2 * 3)
    -6 + 4x = 2x + 2x2 + (-9 + -3x + 6x2)

    Reorder the terms:
    -6 + 4x = -9 + 2x + -3x + 2x2 + 6x2

    Combine like terms: 2x + -3x = -1x
    -6 + 4x = -9 + -1x + 2x2 + 6x2

    Combine like terms: 2x2 + 6x2 = 8x2
    -6 + 4x = -9 + -1x + 8x2

    Solving
    -6 + 4x = -9 + -1x + 8x2

    Reorder the terms:
    -6 + 9 + 4x + x + -8x2 = -9 + -1x + 8x2 + 9 + x + -8x2

    Combine like terms: -6 + 9 = 3
    3 + 4x + x + -8x2 = -9 + -1x + 8x2 + 9 + x + -8x2

    Combine like terms: 4x + x = 5x
    3 + 5x + -8x2 = -9 + -1x + 8x2 + 9 + x + -8x2

    Reorder the terms:
    3 + 5x + -8x2 = -9 + 9 + -1x + x + 8x2 + -8x2

    Combine like terms: -9 + 9 = 0
    3 + 5x + -8x2 = 0 + -1x + x + 8x2 + -8x2
    3 + 5x + -8x2 = -1x + x + 8x2 + -8x2

    Combine like terms: -1x + x = 0
    3 + 5x + -8x2 = 0 + 8x2 + -8x2
    3 + 5x + -8x2 = 8x2 + -8x2

    Combine like terms: 8x2 + -8x2 = 0
    3 + 5x + -8x2 = 0

    Factor a trinomial.
    (1 + -1x)(3 + 8x) = 0


    the factor '(1 + -1x)'

    Simplifying
    1 + -1x = 0

    Solving
    1 + -1x = 0

    Move all terms containing x to the left, all other terms to the right.

    Add '-1' to each side of the equation.
    1 + -1 + -1x = 0 + -1

    Combine like terms: 1 + -1 = 0
    0 + -1x = 0 + -1
    -1x = 0 + -1

    Combine like terms: 0 + -1 = -1
    -1x = -1

    Divide each side by '-1'.
    x = 1

    Simplifying
    x = 1


    the factor '(3 + 8x)'

    Simplifying
    3 + 8x = 0

    Solving
    3 + 8x = 0

    Move all terms containing x to the left, all other terms to the right.

    Add '-3' to each side of the equation.
    3 + -3 + 8x = 0 + -3

    Combine like terms: 3 + -3 = 0
    0 + 8x = 0 + -3
    8x = 0 + -3

    Combine like terms: 0 + -3 = -3
    8x = -3

    Divide each side by '8'.
    x = -0.375

    Simplifying
    x = -0.375

    x = {1, -0.375}
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  6. #6
    ADY
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    Do my workings out figure out ok, i've been looking at it all afternoon but confused,
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  7. #7
    Flow Master
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    Quote Originally Posted by ADY View Post
    Do my workings out figure out ok, i've been looking at it all afternoon but confused,
    It looks like you've correctly smplified the equation to 8x^2 - 5x - 3= 0. Your two solutions are correct, although I prefer to use fractions ...... -3/8 rather than -0.375.

    It's admirable that you're posting all of your work, every single step. Excellent. But such detail can make it difficult to read and follow the plot ..... Perhaps you could post just the key steps. If more detail is needed someone will ask.
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