How do you solve #x^2 + 10x + 5 = 0 # by completing the square?

1 Answer
May 13, 2016

Answer:

#x=-5+-2sqrt(5)#
(see below for completing the squares method of solution)

Explanation:

Given:
#color(white)("XXX")x^2+10x+5=0#

Move the constant to the right side as
#color(white)("XXX")color(blue)(x^2+10x)=-5#

We know that #(x+a)^2=color(red)(x^2+2ax+a^2)#
So if the first two terms of a squared binomial are
#color(white)("XXX")color(red)(x^2+2ax) = color(blue)(x^2+10x)#
then
#color(white)("XXX")color(red)(a)=color(blue)(5)#
and we will need to add
#color(white)("XXX")color(red)(a^2)=color(blue)(25)# (to both sides) to complete the square:

#color(white)("XXX")x^2+10x+25 = -5+25#

Writing as a squared binomial and simplifying the right side:
#color(white)("XXX")(x+5)^2=20#

Taking the square root of both sides:
#color(white)("XXX")x+5=+-2sqrt(5)#

Subtracting #5# from both sides
#color(white)("XXX")x=-5+-2sqrt(5)#