How do you solve using the completing the square method 3x2+3x+2y=0?

1 Answer
Alan P.
Oct 24, 2017

Note that since you have a single equation in two variables, the "solution" can only be written with the inclusion of a variable.

x=−12±√14−23y

Explanation:

Given
XXX3x2+3x+2y=0

Remember that the "target" completed square must be of the form:
XXXm(x−a)2+b(=0)

First we will extract the m factor from the first 2 terms:
XXX3(x2+1x)+2y=0

We also need to remember that the first 2 terms of the expansion of a squared binomial (x−a)2 are x2 and −2ax (and that the third term is a2)

If (x2+1x) are the first 2 terms of an expanded squared binomial then a=−12 and a2=14 will need to be added to these 2 terms to give an expanded completed square.
So this potion of our solution must look like (x2+1x+14)

Notice that this new term 14 is actually being multiplied by the m=3 factor;
so we are actually adding 3×14=34

To avoid changing the value of the expression if we are going to add 34 we will also need to subtract 34;
so we will have
XXX3(x2+1x+14)+2y−34=0

Re-writing with a squared binomial
XXX3(x+12)2+2y−34=0
or in proper vertex form
XXX3(x−(−12))2+2y−34=0

This ends the "completing the square portion";
now onto the solution:
XXX3(x+12)2=34−2y

XXX(x+12)2=14−23y

XXXx+12=±√14−23y

XXXx=−12±√14−23y

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