Quadratic equations – These are the equations which look like
ax2 + bx + c = 0
These equations are asked in the competitive examinations in a set of 5 questions. Enough with the theory, the question in exams look like  In each of the following questions, two equations are given. Solve these equations and give the answer:
A) If x >= y, i.e. x is greater than or equal to y
B) If x > y, i.e. x is greater than y
C) If x <= y, i.e. x is less than or equal to y
D) If x < y, i.e. x less than y
E) If x = y, or no relation can be established between x and y
Ques 1
I. x2 + 5x + 6 = 0
II. y2 +7 y + 12 = 0Older way
x2 + 5x + 6 = 0
We used to solve it as
x2 + 3x + 2x + 6 = 0
x(x + 3) + 2(x + 3) = 0
(x + 3) (x + 2) = 0
Then, we get the two roots which are x = 3, 2
Let’s try all of this but another way.
Consider the same equation:
Consider the same equation:
x2 + 5x + 6 = 0
When we compare this equation to
ax2 + bx + c = 0
we get, a = 1, b = 5, c = 6A way to find the factors
 Start dividing the number formed using “c*a” starting from first prime number i.e. 2. The result gives one factor and the other factor is the number you divided with.
 Add them and then subtract them to see if they give “b”.
 If not divide it with next number on number line i.e. 3 and so on till you get the final factors.
 Don’t worry this method seems lengthy but with a little bit of practice it will get faster.
We have our factors 3 and 2 and now is the time to decide the sign. For that, riddle me this
Therefore, the factors become +3, +2
In the above question, the value of “a” is 1. Therefore, the factors becomeRemember two things before marking them as final answers:
 Change the sign of the factors i.e. – if they are + and + if they are .
 Divide the two factors by “a”.
3/1 = 3 2/1 = 2
And for the last touch, we change the sign. After that, we get the final factors.
For the last touch, we change the signs and divide them with “a”, which in this case is 1.
Final Factors: 3, 2
Let’s take one more example to make process more familiar to you.
x2 – 3x – 10
Final Factors: 2, 5
In each of the following questions, two equations are given. Solve these equations and give the answer:
A) If x >= y, i.e. x is greater than or equal to y
B) If x > y, i.e. x is greater than y
C) If x <= y, i.e. x is less than or equal to y
D) If x < y, i.e. x less than y
E) If x = y, or no relation can be established between x and y
Examples
Ques 1
I. x2 + 6x + 9 = 0
II. 2y2 +3y  5 = 0
x2 + 6x + 9 = 0
Values of x 3, 3
For equation no. 2
2y2 +3y  5 = 0
5/2 = 2.5, 2/2 =1
Values of y: 2.5, 1
We compare every value of x to every value of y and note the results down.
As in all the cases, x < y, we can say with certainty that x < y (final answer).
A) If x >= y, i.e. x is greater than or equal to y
B) If x > y, i.e. x is greater than y
C) If x <= y, i.e. x is less than or equal to y
D) If x < y, i.e. x less than y
E) If x = y, or no relation can be established between x and y
Value of x

Value of y

Conclusion

3 (1st value)

2.5

x < y

3 (1st value)

1

x < y

3 (2nd value)

2.5

x < y

3 (2nd value)

1

x < y

As in all the cases, x < y, we can say with certainty that x < y (final answer).
Practice Questions
In each of the following questions two equations are given. Solve these equations and give answer:A) If x >= y, i.e. x is greater than or equal to y
B) If x > y, i.e. x is greater than y
C) If x <= y, i.e. x is less than or equal to y
D) If x < y, i.e. x less than y
E) If x = y, or no relation can be established between x and y
Q 1:
I. x2 + 5x + 6 = 0
II. y2 +7 y + 12 = 0
Now, we compare
We can say with certainty that x >= y (final answer).
Value of x

Value of y

Conclusion

3

4

x > y

3

3

x = y

2

4

x > y

2

3

x > y

Q 2:
I. x2 + 4x + 4 = 0
II. y2 – 8y + 16 = 0
Now, we compare
Now, we compare
We can say with certainty that x <= y (final answer).
Now, we compare
We can say with certainty that x <= y (final answer).
We can say with certainty that x < y (final answer).
Value of x

Value of y

Conclusion

2 (1st factor)

4 (1st factor)

x < y

2 (1st factor)

4 (2nd factor)

x < y

2 (2nd factor)

4 (1st factor)

x < y

2 (2nd factor)

4 (2nd factor)

x < y

We can say with certainty that x < y (final answer).
Q 3:
I. x2 – 19x + 84 = 0
II. y2 – 25y + 156 = 0
Value of x

Value of y

Conclusion

12

12

x = y

7

12

x < y

12

13

x < y

7

13

x < y

Q 4:
I.4x2 – 8x + 3 = 0
II.2y2 – 7y + 6 = 0
Now, we compare
Value of x

Value of y

Conclusion

1.5

2

x < y

0.5

2

x < y

1.5

1.5

x = y

0.5

1.5

x < y

Q 5:
I. x2 + x – 6 = 0
II.2y2 – 13y + 21 = 0
Now, we compare
Value of x

Value of y

Conclusion

3

3.5

x < y

2

3.5

x < y

3

3

x < y

2

3

x < y

Exercise For Practice
Q 1: I. x2 – x – 6 = 0
II. 2y2 + 13y + 21 = 0
Q 2: .I. x2 + 5x + 6 = 0
II. y2 + 3y + 2 = 0
Q 3: I. x2 + x = 56
II. y2 – 17y + 72= 0
Q 4: I. 3x2 + 17x + 10 = 0
II. 10y2 + 9y + 2 = 0
Q 5: I. 12x2 + 11x + 12 = 10x2 + 22x
II. 13y2 – 18y + 3 = 9y2 – 10y
Q 6: I.3x2 – 19x + 28 = 0
II.5y2 – 18y + 16 = 0
Q 7: I. 6x2 + 5x + 1 = 0
II. 15y2 + 8y + 1 = 0
Q 8: I. x2 + 5x + 6 = 0
II. 4y2 + 24y + 35 = 0
Q 9: I. 2x2 + 5x + 3 = 0
II. y2 + 9y + 14 = 0
Q 10: I. 88x2 – 19x + 1 = 0
II. 132y2 – 23y + 1 = 0