Solving Quadratic Equations

To solve Quadratic Equations with the help of Number Line
1. x² + 4x - 221 = 0 y² - 21y - 196 = 0 
A. X > Y
B. X < Y
C. X ? Y
D. X ? Y
E. X = Y or relation cannot be established
Answer & Explanation
Answer - E. X = Y or relation cannot be established
Explanation :
(x - 13) (x + 17) = 0
x = 13, - 17 y² - 21y + 196 = 0
y = 28, - 7

2. x² - 56x + 255 = 0 y² - 34y + 208 = 0
A. X > Y
B. X < Y
C. X ? Y
D. X ? Y
E. X = Y or relation cannot be established
Answer & Explanation
Answer - E. X = Y or relation cannot be established
Explanation :
x² - 56x + 255 = 0
x = 5, 51 y² - 34y + 208 = 0
y = 8, 26

3. x² + x - 272 = 0 y² - 30y + 224 = 0
A. X > Y
B. X < Y
C. X ? Y
D. X ? Y
E. X = Y or relation cannot be established
Answer & Explanation
Answer - E. X = Y or relation cannot be established
Explanation :
x² + x - 272 = 0
x = 16, -17 y² - 30y + 224 = 0
y = 16, 14

4. x² - 27x + 182 = 0 y²+ 41y - 702 = 0
A. X > Y
B. X < Y
C. X ? Y
D. X ? Y
E. X = Y or relation cannot be established
Answer & Explanation
Answer - C. X ? Y
Explanation :
x² - 27x + 182 = 0
x = 13, 14 y² + 41y - 702 = 0
y = 13, -52

5. x² - 27x + 180 = 0 y² + 20y + 91 = 0
A. X > Y
B. X < Y
C. X ? Y
D. X ? Y
E. X = Y or relation cannot be established
Answer & Explanation
Answer - A. X > Y
Explanation :
x² - 27x + 180 = 0
x = 15, 12 y² + 20y + 91 = 0
y = -13, -7

6. x² + 39x + 324 = 0 6y² - y - 5 = 0
A. X > Y
B. X < Y
C. X ? Y
D. X ? Y
E. X = Y or relation cannot be established
Answer & Explanation
Answer - B. X < Y
Explanation :
x² + 39x + 324 = 0
x = - 27 - 12 6y² - y - 5 = 0
y = 1, -0.8

7. x² + 23x - 420 = 0 y² - 28y + 195 = 0
A. X > Y
B. X < Y
C. X ? Y
D. X ? Y
E. X = Y or relation cannot be established
Answer & Explanation
Answer - B. X < Y
Explanation :
x² + 23x - 420 = 0
x = 12, -35 y² - 28y + 195 = 0
y = 15, 13

8. x² - 20x + 525 = 0 y² + 40y + 391 = 0
A. X > Y
B. X < Y
C. X ? Y
D. X ? Y
E. X = Y or relation cannot be established
Answer & Explanation
Answer - A. X > Y
Explanation :
 x² - 20x - 525 = 0
x = 35, -15 y² + 40y + 391 = 0
y = -23, -17

9. 2x² - 3x - 20 = 0 y² - 9y + 20 = 0
A. X > Y
B. X < Y
C. X ? Y
D. X ? Y
E. X = Y or relation cannot be established
Answer & Explanation
Answer - D. X ? Y
Explanation :
2x² - 3x - 20 = 0
x = - 2.5, 4 y² - 9y + 20 = 0
y = 4, 5

10.x² - 19x + 90 = 0 y² - 16y + 63 = 0
A. X > Y
B. X < Y
C. X ? Y
D. X ? Y
E. X = Y or relation cannot be established
Answer & Explanation
Answer - C. X ? Y
Explanation :
x² - 19x + 90 = 0
x = 10, 9 y² - 16y + 63 = 0
y = 7, 9

11.For which of the following equations the value of X is less than Y(XY) 
I. 3X2+23X + 44 = 0; 3Y2 + 20Y +33 = 0 
II. 3X2+29 X +56 = 0; 2Y2 + 15Y + 25 = 0 
III. 3X2- 16X + 21 = 0; 3Y2 - 28Y + 65 = 0 
1.Only I 
2.Only II 
3.Both I and III 
4.Both II and III 
5.None follow 
Answer & Explanation
Answer -  3. Only III
Explanation :
25/?x + 9/?x = 17?
x 34 = 17x x = 2 21?y/18 = 3/?y
y = 18/7 = 2.57

12.For which of the following equations the value of X is less than or equal to Y (X? Y) 
I. X2- 4X + 3= 0; Y2 - 8Y + 15 = 0 
II. 3X2 - 19X + 28= 0; 4Y2 - 29Y + 45 = 0 
III. x2 - (16)2 = (23)2 - 56; y1/3 - 55 + 376 = (18)2 
1.Only I 
2.Only II 
3.Both I and III 
4.Both II and III 
5.All follow 
Answer & Explanation
Answer -  3. Both I and III
Explanation :
From I, (x-3)(x-1) = 0 X=1,3 (y-5)(y-3) = 0 Y = 5,3
From III, x2-(16)2=(23)2 - 56 x2 = 729
x = ± 27 y1/3 - 55 + 376 = (18)2
y = 33 = 27

13.For which of the following equations the value of X is greater than or equal to Y ( X ? Y) 
I. X2 -3X - 4 = 0; 3Y2 - 10Y+8 = 0
II. ?(x + 6) = ?121 - ?36; y2 + 112 = 473 
III. 5x2- 7x - 6 = 0; 5y2 + 23y + 12 = 0 
1.Only I 
2.Only II 
3.Both I and III 
4.Both II and III 
5.None follow 
Answer & Explanation
Answer -  4. Both II and III
Explanation :
From II, ?(x + 6) = ?121 - ?36 x + 6 = 25; x = 19 y2 + 112 = 473 y2 = 361; y = ± 19
From III, 5x2 - 7x - 6 = 0; x = -3/5, 2 5y2 + 23y + 12 = 0; y = -4, -3/5

14. For which of the following equations the value of X is greater than Y(X>Y)
I. 3X2+23X + 44 = 0; 3Y2 + 20Y +33 = 0 
II. 3X2+29 X +56 = 0; 2Y2 + 15Y + 25 = 0 
III. 3X2- 16X + 21 = 0; 3Y2 - 28Y + 65 = 0 
1.Only I 2.Only II 
3.Both I and III 
4.Both II and III 
5.None follow
Answer & Explanation
Answer -  5. None follow
Explanation :
From I, X = -4, -11/3; Y = -3, -11/3
From II, X=-7, -8/3; Y = -5, -5/2
From III,X= 3,7/3; Y = 5,13/3

15. For which of the following equations the value of X=Y or relationship cannot be established. 
I. 1/x + 1/(x-10) = 8/75; 132/y - 132/(y + 11) = 1 
II. (3x - 2)/y = (3x + 6)/(y + 16); (x + 2)/(y + 4) = (x + 5)/(y + 10)
III. x² - 4x - 21 = 0; y² - 35y + 306 = 0 
1.Only I 
2.Only II 
3.Both I and III 
4.Both II and III 
5.All follow 
Answer & Explanation
Answer -  1.Only I
Explanation :
From I, x = 25, 3.75;
y = - 44, 33

16. For which of the following equations the value of Y is greater than X(Y>X) 
I. 7x + 6y + 4z = 122; 4x + 5y + 3z = 88; 9x + 2y + z = 78 
II. 7x + 6y = 110; 4x + 3y = 59 
III. 2x+5y = 23.5, 5x+ 2y = 22 
1.Only I 
2.Only II 
3.Both I and II 
4.Both II and III
5.All follow 
Answer & Explanation
Answer -  5.All follow
Explanation :
From I, x = 6, y = 8
From II, x = 8; y = 9
From III, x = 3, y = 3.5

17. For which of the following equations the value of X is less than or equal to Y (X? Y) 
I. [48 / x4/7] - [12 / x4/7] = x10/7; y³ + 783 = 999 
II. 9/?x + 19/?x = ?x; y5 - [(28)11/2 /?y] = 0 
III. 6X2 - 7X + 2 = 0; 12Y2 - 7Y+1 = 0 
1. Only I 
2. Only II 
3. Both I and II 
4. Both I and III 
5. All follow 
Answer & Explanation
Answer -  1.Only I
Explanation :
From I, x = ± 6; y = 6

18. For which of the following equations the value of X is greater than or equal to Y (X ? Y) 
I. 3X2 +17X + 10=0; 10Y2 + 9Y+2 = 0 
II. X2 +X - 56 = 0; Y2 - 17Y+72 = 0 
III. X2- 12X + 35= 0; Y2 + Y - 30 = 0 
1. Only I 
2. Only II 
3. Both I and II 
4. Both I and III 
5. None of these 
Answer & Explanation
Answer - 5.None of these
Explanation :
Only III follows, From III, X = 7,5; Y = 5, -6

19. For which of the following equations the value of X is greater than Y(X>Y) 
I. 2X2 + 15X + 25= 0; 3Y2 + 29Y + 56 = 0 
II. 3X2 - 19X + 28= 0; 4Y2 - 29Y + 45 = 0 
III. X2 - 13X + 42= 0; Y2 - 9Y + 20 = 0 
1. Only I 
2. Only II 
3. Only III
4. Both II and III 
5. None of these 
Answer & Explanation
Answer -  3.Only III
Explanation :
From III, X=7,6; Y =5,4

20. For which of the following equations the value of X=Y or relationship cannot be established. 
I. x2 - 18x + 72= 0; 3y2 + 7y + 4 = 0 
II. 2x2 + x - 36 = 0; 2y2 - 13y + 20 = 0 
III. 4X2 - 19X + 12= 0; 3Y2 + 8Y + 4 = 0 
1. Only I
2.Only II 
3. Both I and II 
4. Both II and III 
5. None of these 
Answer & Explanation
Answer -  2.Only II
Explanation :
 From II, x = 4 , -9/2;
y = 4, 5/2

21. (x - 3)(x + 2) = 0 y² - 7y + 12 = 0 
A. X > Y 
B. X < Y 
C. X ? Y 
D. X ? Y 
E. X = Y or relation cannot be established 
Answer & Explanation
Answer - D. X ? Y
Explanation :
(x - 2) (x + 1) = 0
x = 3, -2 y² - 7y + 12 = 0
y = 3, 4

22. 1/x+4 - 1/x-7 = 11/30 1/y-3 + 1/y+5 = 1/3 
A. X > Y 
B. X < Y 
C. X ? Y 
D. X ? Y 
E. X = Y or relation cannot be established 
Answer & Explanation
Answer - E. X = Y or relation cannot be established
Explanation :
1/x+4 - 1/x-7 = 11/30 x = 1, 2 1/y-3 - 1/y+5 = 1/3 y = -3, 7

23. x(35 - x) = 124 y(2y + 3) = 90 
A. X > Y 
B. X < Y C. X ? Y 
D. X ? Y 
E. X = Y or relation cannot be established 
Answer & Explanation
Answer - E. X = Y or relation cannot be established
Explanation :
x(35 - x) = 124 x = 31, 4 y(2y + 3) = 90 y = -7.5, 6

24. x(27 - x) = 182 y(55 - y) = 750 
A. X > Y 
B. X < Y 
C. X ? Y 
D. X ? Y 
E. X = Y or relation cannot be established 
Answer & Explanation
Answer - B. X < Y
Explanation :
x(27 - x) = 182 x = 13, 14 y(55 - y) = 750 y = 25, 30

25. x(x + 1) = 306 y(2y + 1) = 528 
A. X > Y 
B. X < Y 
C. X ? Y 
D. X ? Y 
E. X = Y or relation can not be established 
Answer & Explanation
Answer - E. X = Y or relation cannot be established
Explanation :
x(x + 1) = 306 x = 17, -18 y(2y + 1) = 528 y = 16, -16.5

26. (x + 3)(x + 29) = 360 480/y =[480/(y - 8)] - 3 
A. X > Y 
B. X < Y 
C. X ? Y 
D. X ? Y 
E. X = Y or relation cannot be established 
Answer & Explanation
Answer - E. X = Y or relation cannot be established
Explanation :
(x + 3)(x + 29) = 360 x = 7, -39 480/y = [480/(y - 8)] - 3 y = 40, -32

27. 360/x = 360/(x +5) + 1 y² - 6y - 216 = 0 
A. X > Y 
B. X < Y 
C. X ? Y 
D. X ? Y
E. X = Y or relation cannot be established 
Answer & Explanation
Answer - E. X = Y or relation cannot be established
Explanation :
360/x = 360/(x +5) + 1 x = 40, -45 y² - 6y - 216 = 0 y = -12, 18

28. 2x² + x - 300 = 0 y² - 45y + 324 = 0 
A. X > Y 
B. X < Y 
C. X ? Y 
D. X ? Y 
E. X = Y or relation cannot be established 
Answer & Explanation
Answer - E. X = Y or relation cannot be established
Explanation :
2x² + x - 300 = 0 x = 12, -12.5 y² - 45y + 324 = 0 y = 36, 9

29. (x + 2)(x + 1)=(x -2)(x - 3) (y + 3)(y +2) =(y - 1)(y - 2) 
A. X > Y 
B. X < Y 
C. X ? Y 
D. X ? Y 
E. X = Y or relation cannot be established 
Answer & Explanation
Answer - A. X > Y
Explanation :
 (x + 2)(x + 1) = (x - 2)(x - 3) x = 1/2 = 0.5 (y + 3)(y + 2) = (y - 1)(y - 2) y = - 1/2 = -0.5

30. (x - 2)(x - 3)=(x - 4)(x - 5) (y - 6)(y - 5) =(y - 4)(y - 3) 
A. X > Y 
B. X < Y 
C. X ? Y 
D. X ? Y 
E. X = Y or relation cannot be established
Answer & Explanation
Answer - B. X < Y
Explanation :
(x - 2)(x - 3) = (x - 4)(x - 5) 4x = 14 x = 3.5 (y - 6)(y - 5) = (y - 4)(y - 3) 4y = 18; y = 4.5