Question

How do you solve `x^2 + 8x + 2 = 0` by completing the square?

Answer 1

`x=-4+-sqrt14`

Explanation:

`"express as " x^2+8x=-2`

`"to "color(blue)"complete the square"`

add `(1/2"coefficient of x-term")^2" to both sides"`

`"that is add " (8/2)^2=16" to both sides"`

`rArrx^2+8xcolor(red)(+16)=-2color(red)(+16)`

`rArr(x+4)^2=14`

`color(blue)"take the square root of both sides"`

`sqrt((x+4)^2)=+-sqrt14larr" note plus or minus"`

`rArrx+4=+-sqrt14`

`"subtract 4 from both sides"`

`xcancel(+4)cancel(-4)=+-sqrt14-4`

`rArrx=-4+-sqrt14`

Answer 2

Move +2 to the right side of the equation.
`x^2 + 8x = -2`

Then halve the coefficient of x.
`x^2 + (8/2)x = -2`

Then square that same coefficient.
`x^2 + (8/2)^2x = -2`

Since `(8/2)^2` = `16` we can put that number into the equation.
So, `x^2 + 8x + 16 = -2`

When you find the number to complete the square you must add it to both sides of the equation.
So, `x^2 + 8x + 16 = -2 +16`
= `x^2 + 8x + 16 = 14`

Then factorise `x^2 + 8x + 16` = `(x+4)(x+4)`

Therefore, the answer is: `(x+4)(x+4)=14`