Solving quadratic equation by a special technique-Symmetric
This wonderful method was stumbled upon by me while I was looking for a way to better explain quadratic equation. I discovered the traditional way of solving quadratic equation might not be enough for some people. Better still a more convenient method may be appealing to some people.
So in my search I came across the famous Vedic math’s India where I learn the technique I will be sharing with you.
Hope you haven’t forgotten that a quadratic equation is a polynomial of the second degree. Sometimes you will be given a simple equation that will lead to a quadratic equation.
For example:
X + 1/X = 26/5
X^2 + 1 =26X/5 which implies that 5X^2 +5 = 26X
Bringing 26x to the L.H.S of the equation we have
5X^2 – 26X + 5 = 0 (A clear quadratic equation)
From here now we will want to apply any of the methods we have learn so far to solve this quadratic equation.
Let us apply the formula method which we consider a typical or prototype solution of quadratic equation.
Recall: X=(-b+or-sqrt(b^2-4ac))/2a
Now setting a=5 , b=26 ,c=5
So that:
X=-(26 + (26^2 – (4*5*5))/2 or X=-(26 - (26^2 – (4*5*5))/2
X=5 or X=1/5
Oh yes that’s the solution to that quadratic equation but wait a minute .what if you can avoid all the trouble of going through those steps. What if you can tell the solution by merely looking at the equation. I bet it’s possible and that’s just what I learn on Vedic maths .I am going to teach you exactly what I learn and the way I understand it.
Now let’s take our former example
X + 1/X = 26/5
26/5 is the same as 5 + 1/5 (come on I just divide 26 by 5 which gives you 5 remainder 1 )
X + 1/5 =5 + 1/5
By looking at the symmetric nature of the equation now, we can easily say that
X=5 or X=1/5
Very simple indeed compare to the trouble we go through when we solve the same equation by formula method that we applied to the equation in the first place
Let’s do more of it to better open your understanding.
X + 1/X = 10/3
Oh yes 10/3 is the same as 3+1/3
X + 1/x = 3 + 1/3
By looking at the symmetric nature of the equation now,you can instantly tell the answer.
X=3 or X=1/3 so simple,isn’t it?
Well, I know you may be asking yourself, what if the equation didn’t come in this simple form as the one’s explained above.
Your question is not out of place in fact I am going to show you how to deal with any quadratic equation right away
Take for an example: 5X^2 – 26X + 5 = 0
What you do is divide through first by coefficient of X raise to the power of 2 which is 5 in this case
The equation becomes: X^2 – 26X/5 + 1 = 0
Secondly divide through by X. The equation now becomes:
X – 26/5 + 1/x = 0
Now move -26/5 to R.H.S
X + 1/X = 26/5 (always be mindful of the sign, when positive travel to the other side it turns – and vice versa)
Now you can take it from there. But let’s finish it up together
X + 1/X = 5 + 1/5
Obviously, from the symmetric nature of the equation now,
X=5 or X=1/5
Simple as ABC. What you do when you see a quadratic equation is to try to reduced it to a simple form as we do above so you can solve it quickly.
Looking at symmetric nature of the equation is the technique applied here. It is not limited to just simple equation as above. Let’s consider one more example and we will round up.
(3x + 2) + 1/(3x + 2) = 65/8
Solution:
(3x+2 )+ 1/(3x+2) = 8 + 1/8
You can see from the symmetric nature of the equation that
(3x +2)= 8 and (3x +2) =1/8
I know you can solve that simple equation
X= 2 or X=17/24 that’s what you get when you solve for x
Okay we haven’t take an equation with a negative sign since. Let’s take one
(3x + 2) - 1/(3x + 2) = 63/8
Very similar to the one above but be careful and observe that 63/8 is same as 8-1/8 that’s the trick there. So that:
(3x + 2) - 1/(3x + 2) = 8 - 1/8
Obviously from the symmetric nature of the equation.
(3x +2) = 8 and (3x +2) = -1/8
X= 2 or X= -17
Hope you enjoyed the lesson. Thanks to Vedic math's
Summary:
Reduce the equation to it’s simplest form
Look at the symmetric nature’s of the equation and solve for x
That’s all that is to the technique we learn this time
Exercise: Try the following; leave your answer in the comment box
1. 2x/(5x+1) – (5x +1)/2x = -15/4
2. (4x + 3)/(3x+4) – (3x + 4)/(4x + 3) = 24/5
Answer will be posted after reactions.
