Inequalities

In quadratic Equations We have the following options to choose from :-

i.   X >Y

ii.  Y >X

iii. X >= Y

iv. Y>= X

v.  X = Y or relationship cannot be established.

What is Variables?

An element, a feature, or a factor that is liable to vary or change to find

any equation and represent by x, y, z etc.

 Example: 3x + 27y = 40 is variable

 Example: 2y + 5x = 12 is variable.

What is Constant?

A constant is special number or a real number usually or whose value is

fixed in the context of use is called a constant.

 Example: 5x – 3y + 2z = 24 here is 5, 3, 2, 24 are constant value.

 Example: 4y + 7y – 9z = 53 here is 4, 7, 9, 53 are constant value.

(i) Now, Suppose on solving the quadratic equation we get X = -1, 2 And Y = -2, -3

On putting these values on number line we will see that X lies on the right of Y. Hence, We can say that X>Y or Y<X.

(ii) Now, Suppose if after solving the equation we get X = -1 , 2 and Y= 3,4.

On putting these values on number line we see that Y lies to the right of X without touching X at any point. Hence, We can say that Y>X or X<Y.

(iii) If X= 3,4 and Y= 2,3. Putting this on number line we get X to the right of Y but touching Y at one single point i.e., 3. Hence, X>=Y or Y<=X.

(iv) If X=3,4 and Y=4,5. Putting these values on number line we will see that Y lies to the right of X but touching X at 4. Hence, Y>=X or X<=Y.

(v) Now, Suppose X= 2,4 and Y= 3,5. Putting these values on number line we will see that X touches Y at all the points between 3 and 4 i.e., at 3, 3.1, 3.2, 3.3 etc. upto 4 hence, in this case we cannot relate X and Y and Hence, Will answer as Relationship cannot be established.

Also suppose if y=7 and X= 4,8 then X<y also X>y . Hence, in this case also no relationship can be established.

Example some quadratic equation:
Example 1:

(i) 6X^2 +11X + 3 = 0

(ii) 6X^2 + 10X +4= 0

Answer :
(i) Shortcut tricks :
This equation +6 is coefficient of x² .
+ 11 is coefficient of x
+3 is constant term.
Step 1: we multiply (+6) x (+3) = +18 = 2x3x3
Step 2: we break 2x3x3 in two parts such that addition between them is 11.
So, 2x3x3 = 2 x 9. Also,  9 +2 = 11

      So , +9 and +2 = Sum of is +11 .

      Step 3: Change the sign of both the factors , So +9 = -9 and +2 = -2 .
and divide by coefficient of x² , So we get -9 / 6 = -3 / 2 and -2 / 6 = – 1 / 3 .

      Therefore, X = -3/2 , -1/3

      Similarly Solving (ii),

     6x4= 24 = 2x2x2x3

Break 2x2x2x3 into two parts such that their sum becomes 10.

2x2 and 2x3 are two parts of 24 whose sum is 10.

Now, Change the sign of both the factors  and divide by coefficient of X^2.

So, +4 = > -4/6

And +6 = > -6/6

Y = --2/3 , -1

So, in all X= -1.5, -0.33

And, Y = -0.667, -1

Now, Putting these values on Number line we get to knw that from -0.667 to -1

The values of X and Y coincides .

Hence, as X and Y coincides at more than 1 point,

We can Say X=Y or Relationship between X and Y cannot be established.

Example

6x^2 +11x + 3 = 0

Answer:

Shortcut tricks :

This equation +6 is coefficient of x 2

+ 11 is coefficient of x

+3 is constant term

Step 1: we multiply (+6) x (+3) = +18

Step 2 : we break + 18 in two parts such that addition between them is

11.

So,  +18 = 9 + 2 = 11 . and product of both factors is 18 . So, +9 and +2 =

Sum of is +11

Step 3: Change the sign of both the factors, So +9 = -9 and +2 = -2

and divide by coefficient of x 2  , So we get -9 / 6 = -3 / 2 and -2 / 6 = – 1 /

3 .