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Math Help - Help with reducing quadratic equation to final terms

  1. #1
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    Help with reducing quadratic equation to final terms

    x^4 - 6x^2 + 8 = 0<br />

    x^2=t
    x^4=t^2

    a= 1, b = 6, c = 8

    My answer using the quadratic formula:

    \Rightarrow x = \frac {6 \pm \sqrt {6^2 - 4(1)(8)}}{2(1)}

    \Rightarrow x = \frac {6 \pm \sqrt {4}}{2}

    \Rightarrow x = \frac {3 \pm \sqrt {2}}{}

    How do I reduce this further to get the final answer or is my answer correct? I always get confused. Thanks in advance.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by lilrhino View Post
    x^4 - 6x^2 + 8 = 0<br />

    x^2=t
    x^4=t^2

    a= 1, b = 6, c = 8

    My answer using the quadratic formula:

    \Rightarrow x = \frac {6 \pm \sqrt {6^2 - 4(1)(8)}}{2(1)}

    \Rightarrow x = \frac {6 \pm \sqrt {4}}{2}

    \Rightarrow x = \frac {3} \pm \frac { \sqrt {2}}{2}

    How do I reduce this further to get the final answer. I always get confused. Thanks in advance.
    ok, so this is wrong, since b = -6 not 6. but we don't need the quadratic formula here


    you replaced x^2 with t, so you should have

    t^2 - 6t + 8 = 0 .........now factor

    \Rightarrow (t - 4)(t - 2) = 0

    \Rightarrow t = 4 \mbox { or } t = 2

    But t = x^2

    \Rightarrow x^2 = 4 \mbox { or } x^2 = 2

    \Rightarrow x = \pm 2 \mbox { or } x = \pm \sqrt {2}
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  3. #3
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    Hello, lilrhino!

    You're making it unnecessarily complicated . . . and wrong.


    x^4 - 6x^2 + 8 \:= \:0
    Factor: . (x^2 - 4)(x^2 - 2)\:=\:0


    And we have two equations to solve:

    . . x^2 - 4\:=\:0\quad\Rightarrow\quad x^2 \:=\:4\quad\Rightarrow\quad x \:=\: \pm2

    . . x^2 - 2\:=\:0\quad\Rightarrow\quad x^2\:=\:2\quad\Rightarrow\quad x\:=\:\pm\sqrt{2}

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  4. #4
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    Quote Originally Posted by Jhevon View Post
    ok, so this is wrong, since b = -6 not 6. but we don't need the quadratic formula here


    you replaced x^2 with t, so you should have

    t^2 - 6t + 8 = 0 .........now factor

    \Rightarrow (t - 4)(t - 2) = 0

    \Rightarrow t = 4 \mbox { or } t = 2

    But t = x^2

    \Rightarrow x^2 = 4 \mbox { or } x^2 = 2

    \Rightarrow x = \pm 2 \mbox { or } x = \pm \sqrt {2}
    Thanks Jhevon, I have to watch my signs. I was looking at an example in the book, so I thought I had to use the quadratic formula even though it could be factored as you demonstrated. Thanks again!
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  5. #5
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by lilrhino View Post
    Thanks Jhevon, I have to watch my signs. I was looking at an example in the book, so I thought I had to use the quadratic formula even though it could be factored as you demonstrated. Thanks again!
    yeah, the quadratic formula can be overkill at times. However, i do agree with Soroban that we are making it more complicated than it should be. Generally it helps students to visualize what to do by replacing a squared term with a single variable, but i believe that's wasting writing space. ordinarily, i'd just factor as Soroban did and cut out the middle man, but i didn't really want to shock you. you'd be surprised at some of the things students are shocked with
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  6. #6
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    Quote Originally Posted by Soroban View Post
    Hello, lilrhino!

    You're making it unnecessarily complicated . . . and wrong.

    Factor: . (x^2 - 4)(x^2 - 2)\:=\:0


    And we have two equations to solve:

    . . x^2 - 4\:=\:0\quad\Rightarrow\quad x^2 \:=\:4\quad\Rightarrow\quad x \:=\: \pm2

    . . x^2 - 2\:=\:0\quad\Rightarrow\quad x^2\:=\:2\quad\Rightarrow\quad x\:=\:\pm\sqrt{2}
    I do that often unfortunately. Math is not my strongest subject, so I sometimes make things more complicated that necessary. Thanks for your response.
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  7. #7
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    Quote Originally Posted by Jhevon View Post
    yeah, the quadratic formula can be overkill at times. However, i do agree with Soroban that we are making it more complicated than it should be. Generally it helps students to visualize what to do by replacing a squared term with a single variable, but i believe that's wasting writing space. ordinarily, i'd just factor as Soroban did and cut out the middle man, but i didn't really want to shock you. you'd be surprised at some of the things students are shocked with
    I'm not easily shocked, I may spend more time thinking about it though . It's easier the way Soroban demonstrated it actually, so I'm not shocked. Thanks...
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