Shortcut Trick Approach for Quadratic Equations Problems

 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

Equation I :-    2x²-11x+12=0

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

Equation II :-  2y²-17y+36=0

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

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

X values -5/1 and - 4/1
X = -5, -4

From equation Y² + 5Y + 6 = 0

6×1 = 6
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

X values -15/5 , -3/5

Change signs , X = 3 , 3/5

20×2=40
-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.