QUADRATIC EQUATION: Finding the roots
The most important thing about any polynomial are its roots.
By counting the number of points where the graph cuts or touches the x axis gives us the number of roots possessed by that equation.
In this post we are going know the four ways of finding the roots of a quadratic equation.
Lets understand first what roots mean for any equation in x.
Graphically,
These are points where the graph of that expression in x cut or touch the x axis which is obvious
as theoretically we studied that root is the point where the expression gives y=0 and for any point on graph with y=0 it lies of x axis.
Lets see some examples-
 |
| It has two roots |
 |
| It has no root |
By counting the number of points where the graph cuts or touches the x axis gives us the number of roots possessed by that equation.
Now lets see the ways in which we can solve to find the roots of quadratic equation -
1. Middle Term Splitting method-
This method follows a protocol and it not easy for some equation to apply this however in most places it is applicable.
Lets take an example 2x2 + 3x - 2 = 0
Step 1: Compare this with the standard equation i.e. ax2 + bx + c = 0 in order to know the values of a, b and c.
Step 2: Now find a*c here 2*(-2) = -4
Step 3: Now find two factors of | a*c |(here 4) which can subtracted to give b(here 3)
here those factors will be (4 and 1)
Step 4: Replace (4-1) in place of b(here 3).
Step 5: We get 2x2 + (4-1)x - 2 = 0
Which can easily be factorised to give
(x+2)*(2x-1)=0
Step 6: We get two values of x for which the expression becomes zero which are-
x = -2 and x = 1/2
These are required roots of the equation.
Lets try to solve one more example-
x2 + 3x + 2 = 0
here a*c = 2
hence we will find two factors of 2 which add up to give 3
which are 2 and 1 hence we can 3 by (2+1)
Hence we get (x+1)(x+2) = 0
Hence roots are -1 and -2.
How do we know when to add or when to subtract?
If a*c >0 then add
if a*c <0 then subtract.
Both are evident by above examples.
2. Completing the square-
The most cumbersome method in general is this one, however its application in while solving Mathematics are numerous.
The main aim of this is to make a perfect square out of this quadratic equation.
Lets take an example to understand this-
2x2 + 3x - 2 = 0
Firstly we make the coefficient of x2 = 1 by dividing by a(here 2) .
Now we get-
Now we get the roots 1/2 and -2
3. Sridharacharya Rule or Quadratic Formula-
Easiest way to solve any quadratic -
By just directly substituting the values of a, b and c we can the roots of any quadratic equation.
Actually it is a consequence of method 2.
If we solve for x using the method for ax2 + bx + c = 0 we get the above stated formula.
The proof is as follows-
4. A differeent way.(by Po-Shen Loh)-
This is the most recent way of solving quadratic equation released in 2019.
It is called by different words like "new" , "different" , "unique" way to solve quadratics.
It has also have an protocol which is as follows-
Again lets take an example to understand this-
x2 - 8x + 12 = 0
For this method we need to know that sum of roots is equal to (-b/a) and product is (c/a)
Step 1: Sum of roots = 8
We know that the average of any two numbers is equidistant from any two numbers-
example- 3 + 1 = 4, their average is 2 which equidistant to both the numbers.
Step 2: Assume the roots to be 4+u and 4-u (average of 8 is 4 and u is the distance between the numbers and average).
Step 3: Product = 12
(4+u)(4-u) =12
On solving we get u = +2 or -2.
whatever value of u we take we will ultimately land up with the same two numbers which are 2 and 6 which are indeed the roots of the equation.
For more details, you can visit-
So these were the four ways in which you can solve to find the roots of a quadratic equation.
Stay Tuned for the next post.
For any query comment down there.
Comments
Post a Comment