The technique for solving this is called "completing the square".

Recognize that (x - p)^{2 }= x^{2 }- 2px + p^{2 }

Since the given equation begins with 2x^{2 }, we first need to divide everything by 2 to get:

x^{2 } + 3x/2 + 4 = 0 <----equation 1

From the 1st 2 terms, we can see that p = -3/4. We can now write:

(x - (-3/4))^{2 }=

x^{2 }+ 3x/2 + 9/16

To make this match equation 1, put -4 +9/16 on the right to get:

x^{2 }+ 3x/2 + 9/16 = - 55/16

In the specified format:

**(x + 3/4)**^{2 }** = -55/16**

thus p = -3/4, and q = -55/16

<<Edited to correct the mistake flagged by David W---it looks like the answer is still right>>

Mark H.

11/08/19

David W.

11/08/19