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Can someone please help explain how to complete the square and solve for #7 in the...

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brandih | eNotes Employee

Posted September 18, 2013 at 2:14 PM via web

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Can someone please help explain how to complete the square and solve for #7 in the attached picture?  Thanks!

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ishpiro | Teacher | (Level 1) Associate Educator

Posted September 18, 2013 at 2:40 PM (Answer #2)

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To solve the above equation for x, take the square root of both sides:

`(2x - 1/4)^2 = 129/16`

`2x - 1/4 = +-sqrt(129)/4`

Adding 1/4 to both sides yields `2x = +-sqrt(129)/4 + 1/4 = (1 +-sqrt(129))/4`

Dividing the both sides of the equation by 2 results in two solutions of the quadratic equation:

`x = (1+sqrt(129))/8`  and `x = (1-sqrt(129))/8`

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted September 18, 2013 at 2:23 PM (Answer #1)

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You need to complete the given square, hence, you need to use the following formula, such that:

`a^2 - 2ab + b^2 = (a - b)^2`

You need to notice that the term `b^2` is missing but you can evaluate it because you have the first two terms, `a^2 = 4x^2` and `2ab = x` , such that:

`{(a^2 = 4x^2),(2ab = x):} => {(a = 2x),(2*2x*b = x):} => 4b = 1 => b = 1/4 => b^2 = 1/16`

Hence, you need to complete the square by adding both sides the term `b^2 = 1/16,` such that:

`4x^2 - x + 1/16 = 8 + 1/16`

By the formula `a^2 - 2ab + b^2 = (a - b)^2` , yields:

`4x^2 - x + 1/16 = (2x - 1/4)^2 = (16*8+1)/16`

`(2x - 1/4)^2 = 129/16`

Hence, completing the given square, using the formula `a^2 - 2ab + b^2 = (a - b)^2` , yields the equivalent expression `(2x - 1/4)^2 = 129/16.`

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