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!

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