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How do you solve the quadratic equation by completing the square: $x^2-6x=0$ ?

1 Answer

Jul 13, 2015

$x^2-6 = 0$
$rArr x=0$ or $x=6$
$color(white)("XXXX")$ (by completing the square)

Explanation:

Given $x^2 - 6x = 0$

$color(white)("XXXX")$ If $x^2$ and $-6x$ are the first two terms of a squared binomial:
$color(white)("XXXX")$ $color(white)("XXXX")$ $(x-a)^2 = (x^2 -2ax +a^2)$
$color(white)("XXXX")$ Then (since $-2ax = -6ax$ ) $rArr a = 3$

$color(white)("XXXX")$ and the third term must be $a^2 = 9$

$color(white)("XXXX")$ So we need to add $9$ (to both sides) to "complete the square"
$x^2-6x+9 = 9$

$color(white)("XXXX")$ Rewriting as a squared binomial
$(x-3)^2 = 9$

$color(white)("XXXX")$ Taking the square root of both sides
$x-3 = +-sqrt(9) = +-3$

$color(white)("XXXX")$ Adding 3 to both sides
$x = 6$ or $x = 0$

(Note that, in this case, the solution would be simpler to determine by factoring).

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