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17 June, 07:16

Find the roots of the quadratic equation x2 _ 8x = 9 by completing the square. Show your work.

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  1. 17 June, 10:09
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    The answer is - 1 or 9

    The quadratic equation is ax² + bx + c = 0. But, by completing the square we turn it into: a (x + d) ² + e = 0, where:

    d = b/2a

    e = c - b²/4a

    Our quadratic equation is x² - 8x = 9, which is after rearrangement:

    x² - 8x - 9 = 0

    So, a = 1, b = - 8, c = - 9

    Let's first calculate d and e:

    d = b/2a = - 8 / (2 * 1) = - 8/2 = - 4

    e = c - b²/4a = - 9 - (-8) ² / (4 * 1) = - 9 - 64/4 = - 9 - 16 = - 25

    By completing the square we have:

    a (x + d) ² + e = 0

    1 (x + (-4)) ² + (-25) = 0

    (x - 4) ² - 25 = 0

    (x - 4) ² = 25

    ⇒ x - 4 = √25

    Since √25 can be either - 5 or + 5, then:

    x - 4 = - 5 or x - 4 = 5

    x = - 5 + 4 or x = 5 + 4

    x = - 1 or x = 9
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