How do you complete the square for $x^{2} + 18 x$ $x^2+18x$ ?

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How do you complete the square for $x^{2} + 18 x$ $x^2+18x$ ?

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Answer 1 · 2015-05-31T11:41:39

${(x + 9)}^{2} = x^{2} + 18 x + 81$ $(x+9)^2 = x^2 + 18x +81$

In general,

$a x^{2} + b x + c = a {(x + \frac{b}{2 a})}^{2} + (c - \frac{b^{2}}{4 a})$ $ax^2+bx+c = a(x+b/(2a))^2 + (c - b^2/(4a))$

Notice that the term added to $x$ $x$ is $\frac{b}{2 a}$ $b/(2a)$

Answer 2 · 2015-05-31T11:42:45

For a general form, squared binomial
$XXXXX$ $color(white)("XXXXX")$ ${(x + a)}^{2} = x^{2} + 2 a x + a^{2}$ $(x+a)^2 = x^2+2ax+a^2$

So if $x^{2} + 18 x$ $x^2+18x$ are the first two terms of a squared binomial
$XXXXX$ $color(white)("XXXXX")$ then, in the general form, $a = 9$ $a=9$ and
$XXXXX$ $color(white)("XXXXX")$ $a^{2} = 9^{2} = 81$ $a^2 = 9^2 = 81$

Of course, if we are going to add $9^{2}$ $9^2$ to the expression $x^{2} + 18 x$ $x^2+18x$ we are also going to have to subtract it:
$XXXXX$ $color(white)("XXXXX")$ $x^{2} + 18 x$ $x^2+18x$
$XXXXX$ $color(white)("XXXXX")$ $= x^{2} + 18 x + 9^{2} - 9^{2}$ $= x^2+18xcolor(red)(+9^2) - color(blue)(9^2)$
$XXXXX$ $color(white)("XXXXX")$ $= {(x + 9)}^{2} - 81$ $=color(red)((x+9)^2) color(blue)(- 81)$

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How do you complete the square for $x^{2} + 18 x$ $x^2+18x$ ?

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