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• Hate using Quadratic Formula ? This could help!

Like it or not, we have to solve quadratic equations in our exams - mostly as part of another question, rather than a direct "find the roots" question. Unless we could factorize in one look, most of us go for the traditional quadratic formula which could take time, with its division and square root steps.

Recently I came across another method - which was surprisingly simple, and was hidden in plain sight (at-least for me!)

This method is pioneered by Prof. Po-Shen Loh (CMU)

Here it goes.

For the equation, ax^2 + bx + c = 0, we know sum of the roots = -b/a and product of roots = c/a.

Let's take it one step ahead. consider m is the mid-point of the line connecting roots, we know it's (-b/a)/2.

Let's explain further with a simple example

Solve the roots for x^2 - 10x + 24 = 0

I know, you have factorized it in your head, and know the roots already! But let's keep it aside, and try the new one.

sum = 10
mid point between the roots = 10/2 = 5
which means the roots are at equal distance (say d) from 5 to either side.

(Root1 = 5 - d) ----- d ------- m = 5 ----- d ------- (Root2 = 5 + d)

so roots are (5 -d) and (5 + d)

we know product of roots = 24,
so (5 - d)(5 + d) = 24
25 - d^2 = 24
d^2 = 1

d = 1 (as it is a distance, we take positive value)
so roots are 5 -1 = 4 and 5 + 1 = 6

next one.

Solve the roots for 6x^2 - 19x + 10 = 0

Maybe bit tricky to factorize for a poor mortal, and time consuming with the traditional formula.

Will try with the above method

sum of roots = 19/6
mid point = 19/12
roots are at same distance (say d) from 19/12
so roots are (19/12 - d) and (19/12 + d).

Product = 10/6
(19/12 - d) * (19/12 + d) = 10/6
(19/12)^2 - d^2 = 10/6
d^2 = (19/12)^2 - 10/6 = (19^2 - 10 * 12 * 2)/144 = 121/144 = 11/12 (positive value only, as it's a distance)

roots = (19/12 - 11/12) = 8/12 = 2/3 & (19/12 + 11/12) = 30/12 = 5/2

one more ?

solve the roots for 4x^2 + 3x - 1 = 0

we know the drill :)

sum = -3/4
mid point = -3/8
roots are equidistant (say d) from -3/8
(-3/8 - d) and (-3/8 + d)

product = -1/4
(-3/8 - d) * (-3/8 + d) = -1/4
(-3/8)^2 - d^2 = -1/4
d^2 = 9/64 + 1/4 = (9 + 16)/64 = 25/64
d = 5/8 (take only positive value, as it's a distance)

roots = (-3/8 - 5/8) = -1, & (-3/8 + 5/8) = 1/4

what if roots are imaginary?

solve the roots for x^2 + x + 1 = 0

sum = -1
mid point = -1/2
roots are equidistant (say d) from -1/2
roots = (-1/2 - d) and (-1/2 + d)

product = 1
(-1/2)^2 - d^2 = 1
d^2 = 1/4 - 1 = -3/4
d = i√3/2
roots are -1/2 ± i√3/2

Can you solve the roots for x^2 - 2x + 8/9 = 0 (maybe, even without a pen! :) )

Happy Learning!

posted in Quant Primer
• RE: Quant Boosters - Hemant Malhotra - Set 14

posted in Quant - Boosters
• RE: Question Bank - 100 Arithmetic Questions From Previous CAT Papers (Solved)

@swati-jha Thanks Swati. Corrected the solution.
Mistake was in distance values and is now clean, hopefully :)
Happy learning!

posted in Quant BBQ - Best of Best Questions
• RE: CAT Question Bank - Games & Tournaments

@mahisap I think the answer is:

B
D
34

posted in DILR - Booster
• RE: Discussion Room : Quant

If 76/(4+√7+√11)=p+q√7+r√11+s√77 then p+q+r+s=?

Multiply both sides by (4 + 71/2 + 111/2)

76 = (4 + 71/2 + 111/2)p + 71/2(4 + 71/2 + 111/2)q + 111/2(4 + 71/2 + 111/2)r + 771/2(4 + 71/2 + 111/2)s

Expand out:
76 = 4p + 71/2p + 111/2p + 71/24q + 7q + 771/2q + 111/24r + 771/2r + 11r + 771/24s + 111/27s + 71/211s

Now get everything together:
76 = (4p+7q+11r) + 71/2(p+4q+11s) + 111/2(p+4r+7s) + 771/2(q+r+4s)

Rewrite the LHS:
76 + 0 * 71/2 + 0 * 111/2 + 0 * 771/2 = (4p+7q+11r) + 71/2(p+4q+11s) + 111/2(p+4r+7s) + 771/2(q+r+4s)

So we have the following system of equations:
4p + 7q + 11r = 76
p + 4q + 11s = 0
p + 4r + 7s = 0
q + r + 4s = 0

4 equations in 4 unknowns. Solve.

posted in Discussion Room
• RE: Discussion Room : Quant

@swanandk12 I think it is infinity. As x increases the value of F(x) will keep on increasing. So possible values of F(x) are infinite. Hence the sum of possible values F(x) is infinity. Thoughts?

posted in Discussion Room
• RE: Question Bank - 100 Algebra Questions From Previous CAT Papers (Solved)

@visheshsahni

No one is perfect here yaar and if you find a mistake in a solution and know the right answer, least thing to do is to share the solution. That's how a forum (should) function right ? :)
Corrected the question.

posted in Quant BBQ - Best of Best Questions
• RE: Question Bank - 100 CAT level questions on Time, Speed and Distance topic

t + t/12 = 9 (time taken in the return journey is 1/12th the time taken for the onward journey)
t = 108/13 = 8.30 hours
t + t/6 = 9.68 hours?

posted in Quant BBQ - Best of Best Questions
• CAT 2018 Syllabus and Study Material

CAT (Common Admission Test) is an online exam to test a candidate’s Verbal Ability (VA), Reading Comprehension (RC), Logical Reasoning (LR), Quantitative Ability (QA) and Data Interpretation (DI) skills. CAT is conducted by IIM. Students who aim to get admission in IIMs, IITs and IISc should have an excellent CAT score. They select students for their business administration and management programmes based on this score. CAT 2018 registrations started on 8th Aug 2018. The last date for registration is 19th Sept 2018. Correction window opens in the last week of September 2018. The exam will be conducted on the 25th of Nov 2018. Before starting with the preparation for CAT it is essential to understand the paper pattern and syllabus for CAT

Syllabus for CAT

Verbal Ability and Comprehension Reading (VARC) Syllabus for CAT 2018:
The total number of questions asked in this section are 34 out of which 24 are MCQ type and 10 are Non MCQ type.

Logical Reasoning (LR) syllabus for CAT 2018:
The total number of questions asked in this section are 16 out of which 12 are MCQ type and 2 are Non MCQ type. Key subtopics to focus are:

1. Clocks
2. Binary Logic
3. Blood Relations
4. Assumption
5. Conclusion
6. Calendars
7. Seating Arrangement
8. Logical Sequence
9. Premise
10. Linear & Matrix Arrangement

Data Interpretation (DI) syllabus for CAT 2018:
The total number of questions asked in this section are 16 out of which 12 are MCQ type and 4 are Non MCQ type. Key subtopics to focus are:

1. Data Tables
2. Bar diagrams
3. Graphs
4. Data charts
5. Pie charts
6. Data analysis

Quantitative analysis (QA) syllabus for CAT 2018:
The total number of questions asked in this section are 34 out of which the number of MCQ type are 24-27 and Non-MCQ are 7-10. This is as per the previous 2 years Quant section.

1. LCM and HCF
2. Percentages
3. Profit and Loss
4. Simple and Compound Interest
5. Speed, Time and Distance; Time and Work; Averages etc
6. Ratio and Proportion
7. Geometry (6-8 questions)
8. Algebra (5-7 questions)
9. Venn diagrams; other topic questions (2-3)
10. Pure Math (1-2 questions)
11. Complex Numbers
13. Linear Equations
14. Binomial Theorem
15. Progressions; Logarithm
16. Inequalities
17. Surds and Indices
18. Probability Functions
19. Permutation and Combination
20. Mixtures and Alligations
21. Set Theory
22. Trigonometry
23. Coordinate Geometry

There are many learning Apps like BYJU’S which help you with CAT preparation with required study materials and practice tests like CAT mock test.

For more updates on CAT 2018, students can visit the official website of BYJU’S which is India’s largest education company and largest k-12 learning App in India. Students are also suggested to subscribe to our YouTube channel.

posted in Test Preparation Strategy
• RE: Question Bank - Algebra - Shashank Prabhu, CAT 100 Percentiler

@shreyasnegi13
There are 10 primes in the given range.
4 also satisfies
9 and 25 won't work because we would already have two 3s and 5s respectively in the numerator
So 10 + 1 = 11 values.

posted in Quant BBQ - Best of Best Questions