• 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: 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: 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: Quant Boosters - Hemant Malhotra - Set 14

    Q30) A teacher asks one of her students to divide a 30-digit number by 11. The number consists of six consecutive 1’s, then six consecutive 2’s, and likewise six 3’s, six 4’s and six 7’s in that order from left to right. The student inserts a three-digit number between the last 4 and the first 7 by mistake and finds the resulting number to be divisible by 11. Find the number of possible values of the three-digit number.

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

    let A,B,C,D
    let A got a rs
    B got a * b rupees
    C got a * b * c rupees
    D got a * b * c * d rupees
    a(1+b+bc+bcd)=70
    a * (1+b+bc+bcd) = 2 * 35
    this will be minimum when all values are as close as possible
    1+b+bc+bcd=35
    b(1+c+cd)=34
    so b=2
    1+c+cd=17
    so c(1+d)=16
    so c=4 and d=3
    so a=2,b=2,c=4,d=3
    so 2,4,16,48 these are values

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

    Q29) In a group of four boys, each of them has a certain number of Re. 1 coins with him. Every boy has an amount that is an integral multiple of the amount possessed by every other boy who has an amount less than him. If the total amount with the four boys put together is Rs. 70, and no two boys have the same amount, what is the minimum possible amount (in Rs.) with the boy who has the maximum number of Rs. 1 coins with him?

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

    Q28) How many arrangements can be made in a round table of 8 chairs if 2 of them like to sit opposite to each other ?

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

    1step = 1 way =2^0
    2 step = (1,1) , (2,0) = 2^1 way
    3 step = (1,1,1) , (2,1) ,(1,2) ,(3)=2^2=4 way
    4 step = 2^3= 8 way
    5 step = 2^4 = 16 way
    6 step = 2^5 = 32 way
    now step 7th step will be sum of previous 6 steps= 63
    8th step = sum of previous 6 step = 125

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

    Q27) In how many ways can you climb up 8 steps if the minimum and maximum steps you can take at a time are 1 and 6 respectively?

    posted in Quant - Boosters
  • 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