QUADRATIC EQUATIONS

Directions (Qs. 1 - 25): In each of these questions, two equations (I) and (II) are given. You have to solve both the equations and giveb answer.
 

A) if x > y    B) if x < y
C) if x > y    D) if x < y
E) if x = y or relationship between x and y cannot be established
 

1. I. x² + 5x + 6 = 0
II. y² + 4y + 4 = 0
Sol: x² + 5x + 6 = 0; x₁ = −3; x₂ = −2
y² + 4y + 4 = 0; y₁ = −2; y₂ = −2
Comparison of x and y values
−3 < −2; −3 < −2; −2 = −2; −2 = −2
Hence, x £ y is correct.
 

2. I. x² − 25x + 156 = 0
II. y² + 13y + 42 = 0
Sol: x² − 25x + 156 = 0; x₁ = 13; x₂ = 12
y² + 13y + 42 = 0; y₁ = −7; y₂ = −6
Comparison of x and y values
13 > −7; 13 > −7; 12 > −6; 12 > −6
Hence, x > y is correct.
 

3. I. x² + 9x + 14 = 0
II. y² + 12y − 45 = 0
Sol: x² + 9x + 14 = 0; x₁ = −7; x₂ = −2
y² + 12y − 45 = 0; y₁ = −15; y₂ = 3
Comparison of x and y values
−7 > −15; −7 > −15; −2 < 3; −2 < 3
Hence, the relationship between x and y cannot be established.
 

4. I. x² − 26x + 168 = 0
II. y² + 48y + 575 = 0
Sol: x² − 26x + 168 = 0; x₁ = 12; x₂ = 14
y² + 48y + 575 = 0; y₁ = −23; y₂ = −25
Comparison of x and y values
12 > −23; 12 > −25; 14 > −23; 14 > −25
Hence, x > y is correct.
 

5. I. x² + 44x + 483 = 0
II. y² + 30y + 224 = 0
Sol: x² + 44x + 483 = 0; x₁ = −21; x₂ = −23
y² + 30y + 224 = 0; y₁ = −16; y₂ = −14
Comparison of x and y values
−21 < −16; −21 < −14
−23 < −16; −23 < −14
Hence, x < y is correct.
 

6. I. x² + 23x + 130 = 0
II. y² + 35y + 306 = 0
Sol: x² + 23x + 130 = 0; x₁ = −13; x2 = −10
y² + 35y + 306 = 0; y₁ = −18; y₂ = −17
Comparison of x and y values
−13 > −18; −13 > −17
−10 > −18; −10 > −17
Hence, x > y is correct.
 

7. I. x² + 44x + 483 = 0
II. y² + 30y + 224 = 0
Sol: x² + 44x + 483 = 0; x₁ = −21; x₂ = −23
y² + 30y + 224 = 0; y₁ = −16; y₂ = −14
Comparison of x and y values
−21 < −16; −21 < −14
−23 < −16; −23 < −14
Hence, x < y is correct.
 

8. I. x² + 18x + 77 = 0
II. 3y² − 14y + 15 = 0
Sol: x² + 18x + 77 = 0; x1 = −11; x2 = −7
3y² − 14y + 15 = 0; y1 = 3;
Comparison of x and y values

Hence, x < y is correct.
 

9. I. 17x² − 14x − 3 = 0
II. 2y² − 7y − 60 = 0
Sol: 17x² − 14x − 3 = 0 ; x1 = ; x₂ = 1
2y² − 7y − 60 = 0: y₁ = -4 :

Comparison of x and y values

Hence, the relationship between x and y cannot be established.
 

10. I. x² − x − 12 = 0
II. y² − 17y − 84 = 0
Sol: x² − x − 12 = 0; x₁ = 4; x₂ = −3
y² − 17y − 84 = 0; y₁ = 9; y₂ = −13
Comparison of x and y values
4 < 9; 4 > −13; −3 < 9; −3 > −13
Hence, the relationship between x and y cannot be established.
 

11. I. x² = 144
II. 7y² − 19y + 10 = 0
Sol: x² = 144; x₁ = 12; x₂ = −12
7y² − 19y + 10 = 0 ; y1 = 2 ;
Comparison of x and y values

Hence, the relationship between x and y cannot be established.
 

12. I. x2 + 2x − 35 = 0
II. 2y2 − 7y − 60 = 0
Sol: x² + 2x − 35 = 0; x₁ = −7; x₂ = 5
2y² − 7y − 60 = 0; y1 = −4;  
Comparison of x and y values

Hence, the relationship between x and y cannot be established.

13. I. 4x² − 8x − 5 = 0
II. 6y² + 5y + 1 = 0
Sol: 4x² − 8x − 5 = 0 ;
6y² + 5y + 1 = 0 ;
Comparison of x and y values

Hence, x > y is correct.
 

14. I. x² = 92
II. y² + 8y + 15 = 0
Sol: x² = 92 ; x₁ = 9 ; x₂ = 9
y² + 8y + 15 = 0 ; y₁ = −3 ; y₂ = −5
Comparison of x and y values
9 > −3 ; 9 > −5 ; 9 > −3 ; 9 > −5
Hence, x > y is correct.
 

15. I. 2x² − 3x − 20 = 0
II. y² + 14y − 32 = 0
Sol: 2x² − 3x − 20 = 0 ; x₁ = 4 ; x₂ =
y² + 14y − 32 = 0 ; y₁ = −16 ; y₂ = 2
Comparison of x and y values

Hence, the relationship between x and y cannot be established.
 

16. I. x² + 13x + 42 = 0
II. y² + 31y + 238 = 0
Sol: x² + 13x + 42 = 0 ; x₁ = −6 ; x₂ = −7
y² + 31y + 238 = 0 ;
y₁ = −18 ; y₂ = −19
Comparison of x and y values
−6 > −18 ; −6 > −19
−7 > −18 ; −7 > −19
Hence, x > y is correct.
 

17. I. x² + 215 = 1176
II. y² + 41y + 418 = 0
Sol: x² + 215 = 1176 ; x₁ = 31 ; x₂ = −31
y² + 41y + 418 = 0 ; y₁ = −19; y₂ = −22
Comparison of x and y values
31 > −19; 31 > −22
−31 < −19; −31 < −22
Hence, the relationship between x and y cannot be established.
 

18. I. x² + 37x + 342 = 0
II. y² = 256
Sol: x² + 37x + 342 = 0; x₁ = −18 ; x₂ = −19
y² = 256 ; y₁ = −16 ; y₂ = 16
Comparison of x and y values
−18 < −16 ; −18 < 16
−19 < −16 ; −19 < −16
Hence, x < y is correct.
 

19. I. x² − 21x + 108 = 0
II. 5y² + 23y + 24 = 0
Sol: x² − 21x + 108 = 0 ; x₁ = 9 ; x₂ = 12
5y² + 23y + 24 = 0 ; y₁ = −3 ; y2 =
Comparison of x and y values

Hence, the relationship between x and y cannot be established.
Hence, x > y is correct.
 

20. I. x³ + 340 = 2537
II. y² + 44y + 483 = 0
Sol: x³ + 340 = 2537; x₁ = 13 ; x₂ = 13
y² + 44y + 483 = 0; y₁ = −21 ; y₂ = −23
Comparison of x and y values
13 > −21; 13 > −23
13 > −21; 13 > −23
Hence, x > y is correct.
 

21. I. x² = 4
II. y² + 4y + 4 = 0
Sol: x² = 4; x₁ = −2; x₂ = 2
y² + 4y + 4 = 0; y₁ = −2; y₂ = −2
Comparison of x and y values
−2 = −2 ; −2 = −2 ; 2 > −2 ; 2 > −2
Hence, x > y is correct.
 

22. I. 3x² − 14x + 15 = 0
II. 2y² − 7y − 60 = 0
Sol: 3x² − 14x + 15 = 0 ; x₁ = 3 ;
2y² − 7y − 60 = 0 ; y₁ = −4 ;
Comparison of x and y values

Hence, the relationship between x and y cannot be established.
 

23. I. x² + 29x + 208 = 0
II. y² + 31y + 238 = 0
Sol: x² + 29x + 208 = 0; x₁ = −16 ; x₂ = −13
y_² + 31y + 238 = 0; y₁ = −18 ; y₂ = −19
Comparison of x and y values
−16 > −18 ; −16 > −19
−13 > −18 ; −13 > −19
Hence, x > y is correct
 

24. I. x² − 11x + 30 = 0
II. y² + 14y − 32 = 0
Sol: x² − 11x + 30 = 0; x₁ = 5 ; x₂ = 6
y² + 14y − 32 = 0; y₁ = −16 ; y₂ = 2
Comparison of x and y values
5 > −16 ; 5 > 2 ; 6 > −16 ; 6 > 2
Hence, x > y is correct.
 

25. 
Sol:
x₁ = 16 ; x₂ = 16
y² = 256; y₁ = −16; y₂ = 16
Comparison of x and y values
16 > −16; 16 = 16
16 > −16; 16 = 16
Hence, x > y is correct.

Key 

1-D   2-A   3-E   4-A   5-B   6-A   7-B   8-B   9-E   10-E   11-E   12-E   13-C   14-A   15-E   16-A   17-E   18-B   19-A   20-A   21-C   22-E   23-A  24-A   25-C