a. asked in Science & MathematicsMathematics · 1 decade ago

k^2 - 3K + 5 =0. Determine the...?

Given the fact that

k^2 - 3k + 5 = 0,

determine the value of k^4 - 6k^3 + 9k^2 -7.

Show all steps involved.

Its greatly appreciated.

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  • Anonymous
    1 decade ago
    Favorite Answer

    You could solve k^2 - 3k + 5 = 0 and plug the two solutions into the second expression, but unfortunately the original equation doesn't have any real solutions for k. You can, however, use the original equation to simplify the k^4 expression:

    k^4 - 6k^3 + 9k^2 - 7

    k^2(k^2 - 6k + 9) - 7

    k^2(k^2 - 3k - 3k + 5 + 4) - 7

    k^2([k^2 - 3k + 5] - 3k + 4) - 7

    But k^2 - 3k + 5 = 0, so this is:

    k^2([0] - 3k + 4) - 7

    k^2(-3k + 5 - 1) - 7

    k^2(k^2 - 3k + 5 - k^2 - 1) - 7

    k^2(0 - k^2 - 1) - 7

    -k^4 - k^2 - 7

    -[ (k^2)^2 + k^2 + 7 ]

    -[ (3k-5)^2 + k^2 + 7 ]

    -[ 9k^2 - 30k + 25 + k^2 + 7 ]

    -[ 10k^2 - 30k + 32 ]

    -[ 10k^2 - 30k + 50 - 18 ]

    -[ 10(k^2 - 3k + 5) - 18 ]

    -[ 10(0) - 18 ]

    18

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