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# 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.

### 0 Answers

- Anonymous1 decade agoFavorite 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 ]

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