How do you solve by completing the square: $x^{2} + \frac{5}{2} x = 5$ $x^2+5/2x=5$ ?

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How do you solve by completing the square: $x^{2} + \frac{5}{2} x = 5$ $x^2+5/2x=5$ ?

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Answer 1 · 2015-04-03T20:37:03

To complete the square on the left side we need to add the square of half of the coefficient of the $x$ $x$ term.
(This follows from the observation ${(x + a)}^{2} = x^{2} + 2 a x + a^{2}$ $(x+a)^2 = x^2+2ax+a^2$ )

$x^{2} + \frac{5}{2} x + {(\frac{5}{4})}^{2} = 5 + {(\frac{5}{4})}^{2}$ $x^2+5/2x+ (5/4)^2 = 5 + (5/4)^2$

${(x + \frac{5}{4})}^{2} = \frac{105}{16}$ $(x+5/4)^2 = 105/16$

Taking the square root of both sides
$x + \frac{5}{4} = \pm \frac{\sqrt{105}}{4}$ $x+5/4 = +-sqrt(105)/4$

So
$x = - \frac{5 + \sqrt{105}}{4}$ $x= -(5+sqrt(105))/4$
or
$x = - \frac{5 - \sqrt{105}}{4}$ $x = -(5-sqrt(105))/4$

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How do you solve by completing the square: $x^{2} + \frac{5}{2} x = 5$ $x^2+5/2x=5$ ?

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