Expert: Scott A Wilson - 6/17/2011

Question
I have a question regarding an example given in my text book. It involves the method for solving quadratic equation from India.  I am trying to understand where 12x went.

Here is the problem:
X2 + 3x – 10 = 0
Solved until this point such as:
X2 + 3x = 10
4x^2 + 12x = 40
4x^2 + 12x + 9 = 49
?????????

the final equation and answers are

2x + 3 = +-7, so x = -5,2

my question is how did 4x^2 + 12x +9 = 49 get to 2x + 3 = +-7
i know 4x^2 square root is 2x, i know 9 square root is 3, and i know 49 square root is +-7.  What happens to the 12x, how is that eliminated from the equation, i am hoping you can show or explain step by step how 4x^2 + 12x +9 = 49 becomes 2x + 3 = +-7 specifically how the 12x is factored out.

Thank you for the help!

Answer
There are at least 3 different approaches to this problem.

The equation given is x² + 3x - 10 = 0.

#1) Complieting the square would involve adding 10 at first, which was done, giving x² + 3x = 10.
Then, take 3 and divide it by 2, then square it, giving 1.5² = 2.25.
Adding this to both sides gives x² + 3x + 2.25 = 10 + 2.25.
The lest side is (x+1.5)² and the right side is 12.25, which is 3.5².
We then have (x+1.5)² = 3.5², so x + 1.5 = ±3.5, so x = -1.5 ± 3.5.
This gives x = -5 or x = 2.


#2) The method you used gives 4x² + 12x + 9 = 49.
This is from adding 10 to both sides, multiplying the equation by 4,
and adding 0 more to both sides.  The result factors into (2x+3)² = 7².
That is the same as 2x+3 = ±7.  Taking 2x+3=-7, 2x=-10, so x = -5.  Taking 2x+3=7, 2x=4,x=2.


#3) An even better way is to factor what we have, and that gives (x+5)(x-2) = 0.
That gives x+5 = 0 or x-2 = 0.  The 1st says x = -5 and the 2nd says x = 2.