How to Solve Quadratic Equation? Tips & Tricks
Best Tips and Tricks for How to Solve Quadratic Equation Questions. Quadratic Equation Free PDF. Welcome to the Let’s Study Together online Quantitative Aptitude Study Portal. If you are preparing for Banking & Insurance exams i.e. IBPS PO/Clerk IBPS RRB PO/Clerk, SBI PO/Clerk, RBI, NABARD, SEBI, SSC, Railway, and other government exams, you will come across Quadratic Equation Problems in Quantitative Aptitude Section. Here we are providing you preparation tips article on “How to Solve Quadratic Equation Questions – Tips and Tricks”.
This “How to Solve Quadratic Equation Questions” article is also important for other banking exams such as SBI PO, IBPS Clerk, SBI Clerk, IBPS RRB Officer, IBPS RRB Office Assistant, IBPS SO, SBI SO, and other competitive exams.
How to Solve Quadratic Equation Questions? Tricks & Tips
Shortcut Tricks are very important things in competitive exams. Time is the main factor in competitive exams. If you know how to manage time then you will surely do great in your exam. Most of us skip that part. Few examples of quadratic equations shortcuts are given on this page below. All tricks on How to Solve Quadratic Equations? are provided here. We request all visitors to read all examples carefully. These examples will help you to understand How to Solve Quadratic Equations?
1. Write down the table (given below) before the exam starts, in your rough sheet, for use during the exam, Analyse the (+, -) signs in the problem, and refer to the table of signs,
2. Write down the new (solution) signs, and see if a solution is obtained instantly. If not, then go to step 3.
3. Obtain the two possible values for X & Y, from both the equations,
4. Rank the values and get the solution,
STEP 1
Firstly, when you enter the exam hall, you need to write down the following master table in your rough sheet instantly (only the signs):-
How to Solve Quadratic Equation? – Tips & Tricks
NOTE: The above table should be entered before the exam starts, in the rough sheet. Because, this will save time, and you can refer to it instantly later on.
STEP 2
Try solving the question instantly if possible, by checking whether + or – values can bring us to a conclusion.
See Example No. – 1
x^{2}+7x+12=0, y^{2} – 5y + 6 = 0
The signs of X’s equation are + and +, which means their solution is – and -. Both negative values. (Refer to the table)
The signs of Y’s equation are – and +, which means their solution values are + and +. Both positive values.
Obviously, X’s possible values are both negative… And Y’s possible values are both positive.
Obviously, the solution is X < Y. We found out just by looking at the signs which will take hardly 5 seconds.
So, any question with signs as “+, +” and “-, +” can be instantly solved, (unless its an unsolvable exception like x – 6x + 16 = 0, which can have no real integer as the answer. But this is a very rare occasion).
See another example, Example No. 2:-
x^{2} + 21x – 32 = 0
y^{2}+ 7y – 12 = 0
The question’s signs are + and -, for both X & Y equations. This means X and Y’s values can be both positive or negative, as per master table…. Answer is an instant CND (Can Not Determine). This takes hardly 5 seconds.
Similarly, any question’s 2 equations with all 4 symbols as negative (-, -), will have each variable’s value either a positive or negative, so its again a CND.
STEP 3
If you visit Step 3, it means the question is not an instantly solved in Step 2. Here, we write the 2 possible signs of each variable anyways, as plus or minus, in our rough sheet. We must find the possible values of the variables. Remember, from the 2 symbols (+ or -) derived from the master table, the first symbol connects to the bigger value,and second symbol connects to the smaller value, numerically.
Example 3:-
x^{2} – 12x + 32 = 0
y^{2} – 7y + 12 = 0
The question’s symbols are -, + which means both values will be positive for each variable, x and y.
x^{2} – 4x – 8x + 32 = 0
X (x – 4) – 4 (x – 8) = 0
X = 4, X =8
y^{2}-7y+12=0
y^{2}-4y-3y+12=0
y (y – 4) – 3 (y – 4) = 0
Y= 4, Y = 4
So, X >= Y
This is a very basic example. Let’s move on to fractional example.
Example 4 is given below:-
2x^{2} + 3x – 9 = 0
2y^{2} – 11y + 15 = 0
+, – means X will be (-) or (+).
-, + means Y will be (+) or (+)
We have to solve it as we can’t decide instantly.
Now, due to a variation of a factor attached to squared values, the change in method is that we’ll have to multiply variable less value (9) with factor of the squared variable (2), to get 18 for X’s equation. We factorize 18 as 2 x 3 x 3 and reduce it to 2 values (which add/subtract to 3) which is 6 x 3…. Now, divide both values by 2, because that is factor of x .So, x = -6/2 or +3/2… In other words, x = -3 or 1.5.
Similarly for Y, we know the signs are + and -, which means both values are positive.
And 15 x 2 = 30. (Again, this is done because y^{2} has a multiplying factor of 2)
30 = 2 x 3 x 5. (we’ll have to multiply 2 values of these, to arrive at final 2 values which add upto 11, which are 6 and 5)
11 = 6 + 5. Now, Divide both 6 and 5 by 2 each, to get respective possible values of y.
So, x = -6/2, 5/2
y = 3 or 2.5.
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STEP-4
Rank all the 4 values carefully, considering – as lower values and + as higher, as they normally are in mathematics.
For example,
“-5” will get rank 4,
“-0.8” will get rank 3,
“0.4” will get rank 2,
and “4” will get rank 1.
If there’s no tie up/ draw among the ranks, then the answer is easy to find So, as you can see from the table above, when all 4 values are different, and if the ranks 1 and 2 are together, they make it a senior variable, otherwise, we can’t determine and it becomes a CND.When they are tied up, then we’ll have to work it out on rank to rank comparison basis to nd a final solution, like below. But this we do only in mind:-
Let’s carry Example 3 from the Step 3 mentioned before…
X = -8, -4
Y = -4, -3
X’s ranks would be 4,2
Y’s ranks would be 2,1. (assuming that ranks 2 & 3 are combined to get rank 2)
We compare them by “Best vs Worst” method this. If we ever have to combine a X>Y with X<Y, then its obviously CND. Rest you guys can figure out. Remember to do this Step 4 in mind itself, writing only the ranks on rough sheet and nothing else. Don’t make such tables there when the clock is running out…
Let’s solve Example 4 further from where we left:-
x = -6/2,5/2, y = 3 or 2.5
X’s ranks are=4,2, Y’s ranks are=1,2
We compare best to worst, and solve it in mind itself to get Y >= X.
Now let us check some special cases of inequality.
SPECIAL CASES:-
1. SQUARES – Let us take few examples of non equation values. See example No. 5:-
x^{2} = 1600
y^{2} = 1600
Answer is CND. Because, a square would always mean possibility of a positive as well as negative value,
x = 40, -40
y = 40, -40
Hence, any 2 squares or their variations are always CND, irrespective of the values.
NOTE: Root of a positive value is always positive. If you have to root a negative value, then the question is wrong, and hence, answer is a CND.
See example No. 6:-
x^{2} – 243 = 468
y^{2} + 513 = 1023
Don’t touch the pen. Its a CND. Because, ultimately we’ll get a perfect/ imperfect square, which is always a CND.
2. LINEAR MIXED VARIABLE EQUATIONS
Example 7:-
5x + 4y = 82
4x + 2y = 64
One way to proceed is to equate either the “X”or “Y” part of both equations by multiplication any one equation with a positive integer.
Here, best way would be to double the second equation to form 8x + 4y = 128. We can now substract second equation from rst,
8x + 4y = 128
– 5x + 4y = 82
————————
3x = 36
removing 4y from both sides easily to get X=12,
4y = 82 – 5x = 82 – 60 = 22,
y = 5.5
There’s mostly no positive negative confusion in these cases.So X>Y is right here.
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Here we are providing you a quiz on “Quadratic Equation” based on the latest pattern for your daily practice.
1. In the following questions two equations numbered I and II are given. You have to solve both the equations and choose the correct option.
I. x^{2} +14x -207 = 0
II. y^{2}-33y +266 = 0
A. x < y
B. x > y
C. x ≤ y
D. x ≥ y
E. x = y or relationship cannot be established
2.In the following questions, two equations numbered I and II are given. You have to solve both the equations and choose the correct option.
I. x^{2}-44x +483 = 0
II. 25y^{2} +15y -54 = 0
A. x < y
B. x > y
C. x ≤ y
D. x ≥ y
E. x = y or relationship cannot be established
3.In the following questions, two equations numbered I and II are given. You have to solve both the equations and choose the correct option.
I. x^{2}-31x +184 = 0
II. y^{2}-22y +96 = 0
A. x < y
B. x > y
C. x ≤ y
D. x ≥ y
E. x = y or relationship cannot be established
4.In the following questions, two equations numbered I and II are given. You have to solve both the equations and choose the correct option.
I. 15x^{2} +22x +8 = 0
II. y^{2} +y -462 = 0
A. x < y
B. x > y
C. x ≤ y
D. x ≥ y
E. x = y or relationship cannot be established
5. In the following questions, two equations numbered I and II are given. You have to solve both the equations and choose the correct option.
I. x^{2}-5x -300 = 0
II. y^{2} +46y +528 = 0
A. x < y
B. x > y
C. x ≤ y
D. x ≥ y
E. x = y or relationship cannot be established
6.In the following questions, two equations numbered I and II are given.You have to solve both the equations and choose the correct option.
187x^{2} + 6x – 1 = 0
153y^{2}– 26y + 1 = 0
A. If x ≤ y
B. If x ≥ y
C. If x < y
D. If x > y
E. If x = y or no specific relation cannot be established.
7.In the following questions, two equations numbered I and II are given. You have to solve both the equations and choose the correct option.
I. 10x^{2} +5x -5 = 0
II. y^{2}-16y +15 = 0
A. x < y
B. x > y
C. x ≤ y
D. x ≥ y
E. x = y or relationship cannot be established
8.In the following questions, two equations numbered I and II are given. You have to solve both the equations and choose the correct option.
I. x = √484
II. 7y – 4x = 45
A. x < y
B. x ≥ y
C. x > y
D. x ≤ y
E. x = y or the relationship between x and y cannot be determined
9.In the following questions, two equations numbered I and II are given.You have to solve both the equations and choose the correct option.
I. 3.5x^{2}– 28x + 42 = 0
II. 1.5y^{2} + 3y – 12 = 0
A. x < y
B. x ≥ y
C. x > y
D. x ≤ y
E. x = y or the relationship between x and y cannot be determined
10.In the following questions, two equations numbered I and II are given. You have to solve both the equations and choose the correct option.
I- y^{6} = 729
II- x^{5} = 243
A. x < y
B. x ≥ y
C. x > y
D. x ≤ y
E. x = y or the relationship between x and y cannot be determined
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In PO & SO examinations in the past, we’ve had 5-10 questions on this topic. In the end, remember to practice more and more on this topic, because it can score some badly needed or easily earned scores within seconds. You can also attempt more Quantitative Aptitude quizzes from the below table-
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We hope the given Quantitative Aptitude Topic-wise Questions and Answers PDF will be helpful for their competitive exams like bank exams and other recruitment board exams. So candidates can utilize our Quantitative Aptitude Topic-wise PDF Download for their effective preparation to score more marks in upcoming competitive exams.
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