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How do you complete the square to solve $-x² +6x +9=0$ ?

1 Answer

May 31, 2015

$-x^2+6x+9=0$
is equivalent to
$color(white)("XXXXX")$ $x^2-6x-9=$

to help keep thing simple, move the constant to the right side as
$color(white)("XXXXX")$ $x^2-6x = 9$

in the general form of the squared binomial
$color(white)("XXXXX")$
$(x+a)^2 = x^2+2ax+a^2$
so if $x^2-6x$ are the first two terms of a squared binomial, $a=3$

and to complete the square, we need to add an extra
$color(white)("XXXXX")$ $a^2 = (-3)^2 =9$

Therefore we write
$color(white)("XXXXX")$ $x^2-6x+9 = 9 +9$

$color(white)("XXXXX")$ $(x-3)^2 = 18$

Taking the square root of both sides:
$color(white)("XXXXX")$ $x-3 = +-3sqrt(2)$

and
$color(white)("XXXXX")$ $x= 3+3sqrt(2)$ or $x=3-3sqrt(2)$

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