How do you solve the equation x^2-6x+9=8 by completing the square?

1 Answer
Nov 25, 2016

x=3+-2sqrt2

Explanation:

x^2-6x+9=color(white)a8
color(white)(aaaa^2aa)-9color(white)a-9color(white)(aaa)Subtract 9 from both sides

x^2-color(red)(6)xcolor(white)(aaa)=-1

Divide the coefficient of the x term color(red)(6) by 2 and square the result.

color(red)6/2=color(limegreen)3 => color(limegreen)3^2=color(blue)9

Add color(blue)9 to both sides.

x^2-6x+color(blue)9=-1+color(blue)9

x^2-6x+9=8

Note that you got the exact equation you started with! The next step is to factor the left side into the square of a binomial. The left side of this equation started out in factorable form, but I demonstrated all the steps anyway for future reference.

Factor the left side

(x-3)(x-3)=8color(white)(aaa)

#Rewrite as the square of a binomial.

(x-color(limegreen)3)^2=8color(white)(aaa)Note that color(limegreen)3 is the number you got when you divided the coefficient of the x term by 2.

Square root both side.

sqrt((x-3)^2) =sqrt8

x-3=sqrt(4*2)

x-3= +-2sqrt2
color(white)a+3color(white)(aaaaaaaa)+3

x=3+-2sqrt2