Question

How do you solve using the completing the square method `x^2 +8x+14=0`?

Answer 1

`x = -2.586 " or " -5.414`

Explanation:

NOte that when you square a binomial there is always a particular pattern for the answer.

`(x - 3)^2 = x^2 -6x + 9 " " (ax^2 + bx + c)`
As long as `a = 1`, then there is always the same relationship between b and c.
`"(half of b)"^2`gives `c. " "` check `(-6 ÷ 2)^2 = (-3)^2 = 9`

In `x^2 + 8x + 14 = 0` we would like the left side to be written as

`"(binomial)"^2`

1. Move the constant to the right side.
   `x^2 + 8x " " = -14`

2. Add the missing term to both sides `(b/2)^2`
   `x^2 color(blue)+ 8x + color(red)16 = -14 + color(red)16`

This is the part that is COMPLETING THE SQUARE.
(Add in what is missing to form a perfect square)

1. The left side is the answer to the square of a binomial;
   `(x color(blue)+ 4)^2 = 2`

2. Find the square root of each side.
   `x + 4 = +-sqrt2`

3. Solve for `x` twice, once with `+sqrt2`, once with `-sqrt2`
   `x = +sqrt2 -4 " or " x = -sqrt2 -4`7
   `x = -2.586 " or " -5.414`