QUADRATIC POLYNOMIALS AND EQUATIONS (Type 01) ax² + bx + c = 0 type
Combo Rule (Perfect Quadratic Expression):
We use combination of 2 sutras.
1. Anurupyena (Proportionalty)
2. Adyamadyenantyamantya (1st by 1st and last by last) (explained below):
We are explaining below:
In Anurupyena, we split the middle term (coefficient of x) of quadratic equation in 2 terms such that Proportion/Ratio of coeff of x2 term to 1st coeff of x term = Ratio of 2nd coeff of x term to constant term. That ratio of the 1st 2 coeff is one of the root of equation.
1. Adyamadyenantyamantya
In Adyamadyenantyamantya (Commonly called as Adyamadyena), we divide the first term’s coeff of eq with 1st term of factor obtained above and last term of eq with the last term of the same factor.
Sanskrit Name (For Adyamadyenantyamantya):
आद्यमाद्ये नान्त्यमन्त्येन
English Translation (For Adyamadyenantyamantya):
1st by 1st and last by last
First Factor
x¹ x⁰
m1 a1
( a1x¹ + m1 x⁰ ) = ( a1x + m1)
Second Factor
x¹ x⁰
a/a1 c/m1
( a1x¹ + m1 x⁰ )
= ( a/a1x + c/m1)
Examples:
1x2 + 8x +12 = 0
Anurupyena: Split middle terms coefficient of x, +5 in 2 parts such that product of coefficient of x2 and constant term is equal to the product of parts of b.
1st coeff of x term = Ratio of 2nd coeff of x term to constant term.
[a × c = b ( m × n)]
(1 × 12) = 8 (2 × 6)
1/2 = 6/12
First Factor
x¹ x⁰
1 2
( 1x + 2 )
Adyamadyenantyamantya: Divide the first term’s coeff (2) of eq by 1st term of factor(1) and divide last term of eq (–3) by 2st term of factor (3)So 2nd factor: 2x–1
Second Factor
x¹ x⁰
1/1 12/2 (1x + 2 )
( 1x + 6 )
Similarly,
4x2 + 12x + 5 = (2x+1)(2x+5)
9x2 –15x + 4 = (3x–1)(3x–4)
6x2 + 11x –10 = (2x+5)(3x–2)
Here we come across to another important sutra Gunitasamuccaya Samuccayagunita
Gunitasamuccaya Samuccayagunita
Sanskrit Name:
गुणितसमुच्चयः समुच्चयगुणितः
Commonly called as Gunitasamuccaya.
English Translation:
Product of the sum of the coefficients of the factors = sum of the coefficients in the product.
Example:
4x2 + 12x + 5 = (2x+1)(2x+5)
Sum of the coefficients in the product: 4 + 12 + 5 =21
Product of the sum of the coefficients of the factors: (2+1)(2+5) = 21
Lopana Sthapanabhyam (Subsutra of Adyamadyenantyamantya):
Sanskrit Name:
लोपनस्तापनाभ्याम्
English Translation:
By Alternare Elimination and Retention.
Examples:
1. Factorize
2x2 + 6y2 + 3z2 + 7xy + 11yz + 7zx
We have 3 variables x, y, z.
Applying Lopanasthapana, remove any of the variable. Lets Eliminate z by putting z=0.
Hence the given expression
E = 2x2 + 6y2 + 7xy
= (x+2y) (2x+3y) …
(Combination of Anurupyena & Adyamadyenantyamantya).
Similarly, if y=0, then
E = 2x2 + 3z2 + 7zx
= (x+3z) (2x+z)
As x and 2x are present separately and uniquely.Hence we cay map to get Factors.
E = (x+2y+3x) (2x+3y+z)
2. Factorize
x2 + xy -2y2 + 2xz -5yz -3z2
Eliminating z and factorizing gives
(x-y)(x+2y)
Eliminating y and factorizing gives
(x-z)(x+3z)
But as we see x term is present in all factors so we cannot map with correct terms to get factors. So we need the eliminate 3rd variable as well
Eliminating x and factorizing gives
(-y-z)(2y+3z)
Now very easily we can do mapping and get E = (x-y-z)(x+2y+3z)
Note:
- We can eliminate 2 variables at a time as well. Thus we will have to map on the basis of Single independent term. But it will have few steps more.
- Lopanasthapana method can also be used for calculating HCF of expressions.
- We can also apply Gunitasamuccaya for checking.
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