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* Short Trick Method :*

*Short Trick Method :*

In each of these questions, two equations (I) and (II) are given. You have to solve both the equations and give answer

1) if x > y

2) if x < y

3) if x ≥ y

4) if x ≤ y

5) if x = y or relation cannot be established between 'x' and 'y'.

*I. 2x² - 11x +12=0 , II. 2y²-17y+36=0***2x²-11x+12=0**

*Equation I :-*
1. multiple X² coefficient and constant term = 12 × 2= 24

2. Split X coefficient in order to get its sum is -11 and its multiplication is 24. It's -8 and -3 .

3.Now change signs -8 = 8 and -3 = 3.

4. Divide these numbers with X² coefficient 8/2 = 4 and 3/2 = 1.5

5. X1 = 4 ; X2 = 1.5

**2y²-17y+36=0**

*Equation II :-*
It's also similar to equation I

1. multiple Y² coefficient and constant term = 2×36 =72

2. Split X coefficient in order to get its sum is -17 and its multiplication is 72. It's -9 and -8 .

3.Now change signs -9 = 9 and -8 = 8.

4. Divide these numbers with X² coefficient 9/2 = 4.5 and 8/2 = 4 .

5. Y1 = 4.5 , Y2= 4

**Compare X and Y values**

Case 1 : X1(4)< Y1(4.5)

Case 2 : X1(4) = Y2(4)

Case 3 : X2(1.5) < Y1(4.5)

Case 4 : X2(1.5) < Y2 (4)

From above 4 Cases we can easily find out relation between X and Y

*Answer : X ≤ Y*

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__Examples ::__

__Examples ::__## Ques 1.

1.X² – 7X + 12 = 0 ; Y² – 11Y + 30 = 0

**Solution :**

Multiplication of X² coefficient and constant 12×1 = 12

Spilt X coefficient 7 into its multiplication is 12 addition is - 7, we get -4 and -3

Change signs of of -4 & -3 , now its become 4,3

Divide these numbers with X² coefficient 4/1 , 3/1

**X = 4 , 3**

Multiplication of Y² coefficient and constant 1×30 = 30

Spilt Y coefficient 11 into its multiplication is 30 addition is - 11, we get -6 and -5

Change signs of of -6 & -5 , now its become 6,5

**Y=6,5**

**x < y**

## Ques 2.

X² + 9X + 20 = 0 ; Y² + 5Y + 6 = 0

**Solution :**

From equation X²+9X+20 =0

20 × 1= 20

9 = 5+4

9 = 5+4

X values -5/1 and - 4/1

**X = -5, -4**

From equation Y² + 5Y + 6 = 0

6×1 = 6

5 = 3+2

5 = 3+2

Y values -3/1 and -2/1

**Y = -3,-2**

**x < y**

## Ques 3.

5X² – 18X + 9 = 0 ; 20Y² – 13Y + 2 = 0

**Solution :**

5×9 = 45

- 18 = -15-3

- 18 = -15-3

X values -15/5 , -3/5

Change signs ,

**X = 3 , 3/5**

20×2=40

-13 = -8-5

-13 = -8-5

Y values -8/20 , -5/20

Change signs =

**Y = 8/20 , 5/20**

**x > y**

**Note:**By solving more problems using this method , You can able to solve these type of problems without pen and paper.