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Determine the variation function π of β for π of π₯ is equal to negative π₯ squared plus ππ₯ plus 17 at π₯ is equal to negative one.
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Additionally, find π if π of four over nine is 11 over six.
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Weβre given the function π of π₯ is negative π₯ squared plus ππ₯ plus 17 and asked to find the variation function for this at π₯ is negative one.
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To do this, we recall that for a function π of π₯ at π₯ is equal to πΌ, the variation function π of β is π of πΌ plus β minus π of πΌ, where β is the change in π₯ from π₯ is equal to πΌ.
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Once we found our function π of β, weβll then substitute β is four over nine to find the value of π in our function π.
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So weβre given π of π₯ is negative π₯ squared plus ππ₯ plus 17.
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With π₯ is negative one, this is equal to our πΌ.
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And substituting πΌ is negative one, we have π of β is π of negative one plus β minus π of negative one.
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And now evaluating our function π at π₯ is equal to negative one plus β, we have negative negative one plus β squared plus π times negative one plus β plus 17.
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That is, π of negative one plus β is equal to negative one minus two β plus β squared minus π plus πβ plus 17.
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And multiplying our parentheses by negative one and collecting like terms, we have negative β squared plus β times π plus two plus 16 minus π.
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Now, evaluating π at π₯ is equal to negative one, we have negative negative one squared plus π times negative one plus 17, that is, negative one minus π plus 17, which is 16 minus π.
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And substituting our two results into our function π of β, we see that 16 minus π minus 16 minus π is equal to zero so that π of β is equal to negative β squared plus β times π plus two.
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Our variation function for π of π₯ is equal to negative π₯ squared plus ππ₯ plus 17 at π₯ is negative one is equal to π of β is negative β squared plus β times π plus two.
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Now weβre given that π of four over nine is equal to 11 over six.
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And this means that if we substitute β is equal to four over nine into π of β, this should equal 11 over six.
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And this means that negative four over nine squared plus four over nine times π plus two is 11 over six.
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And weβre going to use this to find the value of π.
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Evaluating this gives us negative 16 over 81 plus four π over nine plus eight over nine is equal to 11 over six.
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And now if we add 16 over 81 and subtract eight over nine from both sides and multiplying both sides by nine over four, we can isolate π on the left-hand side.
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Canceling through our parentheses, we then have π is equal to 33 over eight plus four over nine minus two, which evaluates to 2.5694 and so on.
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So with our variation function π of β is negative β squared plus β times π plus two, we have a value of π equal to 2.57 to two decimal places.