Question

How do you solve `4x^2 +12x=29` by completing the square?

Answer 1

Please see the explanation.

Explanation:

Divide both sides by 4:

`x^2 + 3x = 29/4`

Add `a^2` to both sides:

`x^2 + 3x + a^2= 29/4 + a^2" [1]"`

Using the pattern `(x+a)^2 = x^2 + 2ax + a^2`, set the middle term of the pattern equal to the middle term on the left of equation [1]:

`2ax = 3x`

Divide both sides of the equation 2x:

`a = 3/2`

Substitute `3/2` for "a" on both sides of equation [1]:

`x^2 + 3x + (3/2)^2= 29/4 + (3/2)^2" [2]"`

We know that the left side of equation [2] is a perfect square, therefore, we can substitute the left side of the pattern with `a = 3/2` into equation [2]:

`(x + 3/2)^2= 29/4 + (3/2)^2" [3]"`

Simplify the right side of equation [3]:

`(x + 3/2)^2= 38/4" [4]"`

Use the square root on both sides:

`x + 3/2= +-sqrt38/2" [5]"`

Subtract `3/2` form both sides:

`x = -3/2 +-sqrt38/2" [6]"`

Split equation [6] into two equations:

`x = -(3 +sqrt38)/2 and x = -(3 -sqrt38)/2`