Question

How do you solve `x^2 + 2x = 3` by completing the square?

Answer 1

Because the sign of the x term is positive, we use the pattern:
`(x+a)^2=x^2+2ax+a^2" [1]"`

Explanation:

Given: `x^2 + 2x = 3`

Add `a^2` to both sides:

`x^2 + 2x + a^2= 3 + a^2" [2]"`

Please observe that the left side of equation [2] now resembles the right side of equation [1]. This means that we can set the middle term of equation [1] equal to the middle term of equation [2], to find the value of "a":

`2ax = 2x`

`a = 1`

Substitute 1 for "a" in equation [2]:

`x^2 + 2x + 1^2= 3 + 1^2" [3]"`

Because we have completed the square, the left side of equation [3] collapses into a square with `a =1` and the right side becomes a single constant:

`(x + 1)^2 = 4" [4]"`

Perform the square root operation on both sides:

`x + 1 = +-2" [5]"`

Subtract 1 from both sides:

`x = -1 +-2" [6]"`

`x = 1 and x = -3`