Class 10th maths Quadratic Equations

 Class 10th maths 

Quadratic Equations

Solve the following quadratic equations: 

1. x2 + 11x + 30 = 0               

2. x2 + 18x + 32 = 0 

3. x2 + 7x – 18 = 0                 

4. x2 + 5x – 6 = 0 

5. x2 – 4y + 3 = 0                   

6. x2 – 21x + 108 = 0 

7. x2 – 11x – 80 = 0               

8. x2 – x – 156 = 0 

9. x2 – 32z – 105 = 0             

10. 40 + 3x – x2 = 0 

11. 6 – x – x2 = 0                    

12. 7x2 + 49x + 84 = 0 

13. m2 + 17mn – 84n2 = 0     

14. 5x2 + 16x + 3 = 0 

15. 6x2 + 17x +12 = 0           

16. 9x2 + 18x + 8 = 0 

17. 14x2 + 9x + 1 = 0 

18. 2x2 + 3x – 90 = 0 

19. 2x2 + 11x – 21 = 0 

20. 3x2 – 14x + 8 = 0 

21. 18x2 + 3x – 10 = 0 

22. 15x2 + 2x – 8 = 0 

23. 6x2 + 11x – 10 = 0

Solve the following quadratic equation (if they exist) by the method of completing the square:

1. 8x2 -22x-21= 0

2. 2x2 - x + 8 = 0

3. 4 3x2 +5x-2 3 = 0
4.      2x2 +7x+5 2 = 0 
5. 9x2 -15x+6 = 0
6. 2x2 -5x+3= 0 
7. 4x2 +3x+5 = 0 
8. 5x2 -6x-2 = 0
9. 4x2 +4bx -(a2 -b2) = 0 
10. a2x2 -3abx+2b2 = 0 
11. x2 -( 3 +1)x+ 3 = 0 
12. x2 -4ax+4a2 -b2 = 0 
13. x2 -( 2 +1)x+ 2 = 0

14.  3x2 +10x+7 3 = 0

15.     2x2 -3x-2 2 = 0

 16. 4x2 +4 3x+3= 0

 17. 2x2 + x+4 = 0 

18. 2x2 + x-4 = 0 

19. 3x2 +11x+10 = 0 

20. 2x2 -7x+3= 0 

21. 5x2 -19x+17 = 0 

22. 2x2 + x-6 = 0 

23. 2x2 -9x+7 = 0 

24. 6x2 +7x-10 = 0 

25. x2 -4 2x+6 = 0

Show that each of the following equations has real roots, and solve each by using the quadratic formula:

1. 9x2 +7x-2 = 0

2. x2 +6x+6 = 0 

3. 2x2 +5 3x +6 = 0  

4. 36x2 -12ax+(a2 -b2 ) = 0

 5. a2b2x2 -(4b4 -3a4 )x-12a2b2 = 0

 6. (a+b)2 x2 -4abx-(a -b)2 = 0
7. 4x2 -2(a2 +b2)x+a2b2 = 0 

8. 9x2 -9(a +b)x+(2a2 +5ab+2b2) = 0

9. 4x2 -4a2x+(a4 -b4) = 0

10.     3x2 +11x+6 3 = 0 

11. 4 3x2 +5x-2 3 = 0                                         

12. 3 7x2 +4x- 7 = 0

13.     7x2 -6x-13 7 = 0

14. 4 6x2 -13x-2 6 = 0 

15. x2 -(1+ 2)x+ 2 = 0

16. 2x2 +5 3x +6 = 0

17. x2 -2x+1= 0

18. 3x2 +2 5x-5 = 0

19. 3a2x2 +8abx +4b2 = 0,a ¹ 0

20. 2x2 -2 6x +3= 0

21. 3x2 -2x+2 = 0

22.     3x2 +10x-8 3 = 0

23. x2 + x+2 = 0 

24. 16x2 = 24x+1 

25.25x2 +20x+7 = 0

26. 6x2 + x-2 = 0

27. x2 +5x+5 = 0

28. p2x2 +(p2 -q2)x-q2 = 0

29. abx2 +(b2 -ac)x -bc = 0

 30. x2 -2ax +(a2 -b2) = 0

 31. 12abx2 -(9a2 -8b2)x -6ab = 0

 32. 24x2 – 41x + 12 = 0

33. 2x2 – 7x – 15 = 0 

34. 6x2 + 11x – 10 = 0 

35. 10x2 – 9x – 7 = 0

36. x2 – x – 156 = 0

37. z2 – 32z – 105 = 0

38. 40 + 3x – x2 = 0 

39. 6 – x – x2 = 0

40. 7x2 + 49x + 84 = 0

 

 

NATURE OF ROOTS  

1. Find the value of k for which the quadratic equation 2x2 + kx + 3 = 0 has two real equal  roots

2. Find the value of k for which the quadratic equation kx(x – 3) + 9 = 0 has two real equal roots.

3. Find the value of k for which the quadratic equation 4x2 – 3kx + 1 = 0 has two real equal roots..

4. If –4 is a root of the equation x2 + px – 4 = 0 and the equation x2 + px +q = 0 has equal roots, find the value of p and q.

5. If –5 is a root of the equation 2x2 + px – 15 = 0 and the equation p(x2 + x) +k = 0 has equal roots, find the value of k.

6. Find the value of k for which the quadratic equation (k – 12)x2 + 2(k – 12)x + 2 = 0 has two real equal roots..

7. Find the value of k for which the quadratic equation k2x2 – 2(k – 1)x + 4 = 0 has two real equal roots..

8. If the roots of the equation (a – b)x2 + (b – c)x + (c – a) = 0 are equal, prove that b + c = 2a.

9. Prove that both the roots of the equation (x – a)(x – b) + (x – b)(x – c)+ (x – c)(x – a) = 0 are real but they are equal only when a = b = c.

10. Find the positive value of k for which the equation x2 + kx +64 = 0 and x2 – 8x +k = 0 will have real roots.

11. Find the value of k for which the quadratic equation kx2 – 6x – 2 = 0 has two real roots.

12. Find the value of k for which the quadratic equation 3x2 + 2x + k= 0 has two real roots.

13. Find the value of k for which the quadratic equation 2x2 + kx + 2 = 0 has two real roots.

14. Show that the equation 3x2 + 7x + 8 = 0 is not true for any real value of x.

15. Show that the equation 2(a2 + b2)x2 + 2(a + b)x + 1 = 0 has no real roots, when a ¹ b.

16. Find the value of k for which the quadratic equation kx2+ 2x +1 = 0 has two real and distinct roots.

17. Find the value of p for which the quadratic equation 2x2 + px +8= 0 has two real and distinct roots.

18. If the equation (1 + m2)x2 + 2mcx + (c2 – a2) = 0 has equal roots, prove that c2 = a2(1 + m2).

19. If the roots of the equation (c2 – ab)x2 – 2(a2 – bc)x + (b2 – ac) = 0 are real and equal, show that either a = 0 or (a3 + b3 + c3) = 3abc.

20. Find the value of k for which the quadratic equation 9x2 + 8kx + 16 = 0 has two real equal roots.

21. Find the value of k for which the quadratic equation (k + 4)x2 + (k+1)x + 1 = 0 has two real equal roots.

22. Prove that the equation x2(a2 + b2) + 2x(ac + bd) + (c2 + d2) = 0 has no real root, if ad ¹ bc.

23. If the roots of the equation x2 + 2cx + ab = 0 are real unequal, prove that the equation x2 – 2(a + b) + a2 + b2 + 2c2 = 0 has no real roots.

24. Find the positive values of k for which the equation x2 + kx + 64 = 0 and x2 – 8x + k = 0 will both have real roots.

25. Find the value of k for which the quadratic equation (k + 4)x2 + (k + 1)x + 1 = 0 has equal roots.

26. Find the value of k for which the quadratic equation x2 – 2(k + 1)x + k2 = 0 has real and equal roots.

27. Find the value of k for which the quadratic equation k2x2 – 2(2k – 1)x + 4 = 0 has real and equal roots.

28. Find the value of k for which the quadratic equation (k + 1)x2 – 2(k – 1)x + 1 = 0 has real and equal roots.

29. Find the value of k for which the quadratic equation (4 – k)x2 + (2k + 4)x + (8k + 1) = 0 has real and equal roots.

30. Find the value of k for which the quadratic equation (2k + 1)x2 + 2(k + 3)x + (k + 5) = 0 has real and equal roots.