As all of you know that quadratic equations are one of the most important topics of banking exam and many students fail to attempt these questions because either they find it too lengthy or they are unable to derive a conclusion through equations. Today I will share some tips and tricks to solve quadratic equations I am sure this will make your life very easy and you will be able to solve equations just in few seconds.
Let us take a case:
For each of the given equation determine whether :
- X>Y 2. X<Y 3. X≥Y 4. X≤Y 5. X=Y or No relation can be established between ‘X’ and ‘Y’.
- 6×2 – 19x + 15 = 0
II. 10y2 – 29y + 21 = 0
Solution: (Textual method)
Hence, x₁=3/2 & x₂=5/3
Hence, y₁=3/2 & y₂=7/5
We have studied this in class 8 or 9 thus finding roots of an equation is an easy task but we make mistakes when we have to compare two variables. Many students think that determining relationship between ‘x’ & ‘y’ means comparing x₁ and x₂ with y₁ and y₂ but this gives you a wrong result.
Rule 1 : To determine relationship between ‘X’ and ‘Y’ , you need to compare x₁ with y₁, y₂ and x₂ with y₁, y₂ or vice versa.
Look at the above roots:
X₁=3/2 or 1.5 , X₂=5/3 or 1.666
Y₁= 3/2 or 1.5, Y₂= 7/5 or 1.4
Here X₁=Y₁, X₁>Y₂ ; X₂>Y₁, X₂>Y₂ , Thus X≥Y (Option 3)
Trick to solve faster :
To find out roots of equation follow below mentioned steps:-
- Multiply coefficient of x² and last number (6*15=90)
2. Factorize 90 in such a way that you get coefficient of x by adding the two factors
i.e 90= (9*10) , 9+10 also gives you 19(coefficient of x)
3. Divide the two factors of 90 by coefficient of X² i.e 9/6 and 10/6, This will give you x₁=3/2(1.5) & x₂=5/3(1.666)
4. If the coefficient of ‘X or Y’ are negative their roots would be positive and if their coefficient are positive, roots would be negative.
4. Follow same steps for 2nd equation and compare the variables to obtain result.
Let us take another case:
This is an important question and every banking exam surely has one question of this type (Case-Relationship between X & Y cannot be established).
Calculating roots 8*21= 168 , 168= 24*7 , x₁= -(24/8) ; x₂= -(7/8)
5*-(36) = -(180) , -(180)= -20*9, y₁= -(20/5) ; y₂= (9/5)
X₁= -3 ; X₂= -0.87
Y₁= -4 ; Y₂= 1.8
Here, ( X₁ > Y₁ , X₁<Y₂ ); ( X₂>Y₂, X₂<Y₂ ) Hence, relationship between X & Y cannot be established(Refer Rule 1)
I will provide you with some tips so that you never make mistake in this case.
Look at the Number line (Figure) below, here roots of X are coming between roots of Y. This is a case of sandwich, Thus, No relationship can be established between X & Y.
Point to remember:- whenever any of the roots of either ‘X’ or ‘Y’ come between roots of ‘X’ or ‘Y’. Relationship between the variables cannot be determined.
Let us take one more question:
Calculating the roots : X= ±12 , Y= 13
Comparing the variables: (X₁= 12, X₂= -12) ; Y= 13
X₁<Y , X₂<Y Thus, X<Y.
Point to remember in this case : X²= 144, Any such equation will always give you two roots (±12) as (12²=144)
& (-12²=144) square of both the roots give the same value.
Important points to keep in mind while marking your answer:
- X>Y : When each of the values of X are greater than each of the values of Y.
2. X<Y : When each of the values of X are lesser than each of the values of Y.
3. X≥Y : When one of the value of X is greater than each of the values of Y & one of the values of X is equal to either one or all of the values of Y.
4. X≤Y : When one of the value of Y is lesser then each of the values of Y & one of the values of X is equal to either one or all of the values of Y.
5. X=Y or relationship between X and Y cannot be established : Whenever there is a case of sandwich, this option follows.
I hope this post was helpful for you guys, practice more questions to gain perfection in solving quadratic equations. All the best for your exams !
(Written by Yeshu)