रविवार, जून 18, 2017

Factoring of quadratic expression - 2

factoring of quadratic exprssions - 1 me maine aapko x²+bx+c type k question ka factor karna shikhaya tha.

Factoring of quadratic expression - 2

Quadratic expression k 2nd type k question ka factor karne k liye maine niche working rule diya hai. Aap step-by-step follow kariye.

Working rule -:

Suppose karte hai ki, ax²+bx+c ek quadratic expression hai to
Step-1. Sabse pahle a aur c ka multiply ( a×c ) kariye.

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Step-2. Agar a×c ki value positive ho. Means a×c = +ac to iss multiplication result k aise gununkhand ( factor ) karenge ki unka sum ( add) "b" k equal ho.
For example -: Maan lete hai ki 4x²+12x+5 ek quadratic expression hai. to a = 4 , b = 12 aur c = 5 aur 
a×c = 4×5 = 20 
Ab hume 20 k aise factor karna hai ki , unn factor ko add karne par "b" equal ho. Means 12 k equal ho.
Factor of a×c = 20 = 10×2
 aur            b = 12 = 10+2
Step-3. Agar a×c ki value  negative  ho.Means a×c = -ac to iss multiplication result k aise gununkhand ( factor ) karenge ki unka subtraction (minus) "b" k equal ho.
For example -:
 Maan lete hai ki 2x²+9x-5 ek quadratic expression hai. to  a = 2 , b = 9 aur c = -5 aur 
a×c = 2×(-5) = -10
Ab hume 10 k aise factor karna hai ki , unn factor ko subtration karne par "b" equal ho. Means 10 k equal ho.
Factor of a×c = -10 = 10×(-1)
 aur            b = 9 = 10-1      



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Step-4.  Ab aap "b" ki value quadratic expression ax²+bx+c me rakh kark 2 parts me divide kark factor ko find kar sakte hai.
Aayiye ab kuch question ka factor karte hai .

Some questions of quadratic expression with factors


Q-1.  49x²+14x+1 ka gununkhand kijiye.

Sol.-: Diya hai , 49x²+14x+1 quadratic expression hai.
a = 49 , b = 14 aur  c = 1
Ab "a×c" = 49×1 = 49
Factor of  "a×c" = 49 = 7×7
aur              " b " = 14 = 7+7
49x²+14x+1 = 49x²+(7+7)x+1
                     = 49x²+7x+7x+1
                     = (49x²+7x)+(7x+1)
                     = 7x(7x+1)+1(7x+1)
                     = (7x+1)(7x+1)     Ans.

Q-2.   9x²-12x+4 ka factors ( gununkhand ) kijiye.

Sol.-:      a = 9 , b = -12 aur  c = 4

Ab "a×c" = 9×4 = 36
Factor of  "a×c" = 36 = 6×6
aur              " b " = 12 = 6+6
9x²-12x+4 = 9x²-(6+6)x+4
                     = 9x²-6x-6x+4
                     = (9x-6x)-(6x-4)
                     = 3x(3x-2)-2(3x-2)
                     = (3x-2)(3x-2)     Ans.

Q-3.  6x²-31x+35 gununkhand kariye. 

Sol.-:      a = 6 , b = -31 aur  c = 35

Ab "a×c" = 6×35 = 210
Factor of  "a×c" = 315 = 21×10
aur              " b " = 31 = 21+10
6x²-31x+35 = 6x²-(21+10)x+35
                     = 6x²-21x-10x+35
                     = (6x-21x)-(10x-35)
                     = 3x(2x-7)-5(2x-7)
                     = (2x-7)(3x-5)     Ans.

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Q-4.  14x²-29x+15 gununkhand gyat kijiye.

Sol.-:
a = 14 , b = -29 aur  c = 15
Ab "a×c" = 14×15 = 210
Factor of  "a×c" = 210 = 14×15
aur              " b " = 29 = 14+15
14x²-29x+15 = 14x²-(14+15)x+15
                     = 14x²-14x-15x+15
                     = (14x²-14x)-(15x-15)
                     = 14x(x-1)-15(x-1)
                     = (x-1)(14x+15)     Ans.

Q-5.  18x²+53x-35 factors kare .

Sol.-:

a = 18 , b = 53 aur  c = -35
Ab "a×c" = 18×(-35) = -630
Factor of  "a×c" = -630 = 10×63
aur              " b " = 53 = 63-10
18x²+53x-35 = 18x²+(63-10)x-35
                     = 18x²+63x-10x-35
                     = (18x²+63x)-(10x+35)
                     = 9x(2x+7)-5(2x+7)
                     = (2x+7)(9x-5)     Ans.


Q-6.  4-x-5x² find factors ?

Sol.-: sabse pahle iss question ko standard form me change karenge.
-5x²-x+4 usk baad
a = -5 , b = -1 aur  c = 4
Ab "a×c" = -5×4 = -20
Factor of  "a×c" = -20 = 5×(-4)
aur              " b " = 1 = 5-4
-5x²-x+4 = -5x²-(5-4)x+4
                     = -5x²-5x+4x+4
                     = (-5x²-5x)+(4x+4)
                     = -5x(x+1)+4(x+1)
                     = (x+1)(4-5x)     Ans.

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Q-7.  20-9x-20x² factors kya hoga ?

Sol.-: Sabse pahle iss question ko standard form me change karenge.
-20x²-9x+20  Ab
a = -20 , b = -9 aur  c = 20
Ab "a×c" = -20×20 = -400
Factor of  "a×c" = -400 = 16×(-25)
aur              " b " = 9 = 25 - 16
-20x²-9x+20 = -20x²-(25-16)x+20
                     = -20x²-25x+16x+20
                     = (-20x²-25x)+(16x+20)
                     = -5x(4x+5)+4(4x+5)
                     = (4x+5)(4-5x)     Ans.

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Q-8.  10x²+3x-4 factors kya hoga ?

Sol.-:
a = 10 , b = 3 aur  c = -4
Ab "a×c" = 10×(-4) = -40
Factor of  "a×c" = -40 = 8×(-5)
aur              " b " = 3 = 8-5
10x²+3x-4 = 10x²+(8-5)x-4
                     = 10x²+8x-5x-4
                     = (10x²+8x)-(5x+4)
                     = 2x(5x+4)-1(5x+4)
                     = (5x+1)(2x-1)     Ans.

Q-9.   7x²-48x-7 ka gununkhand kariye.

Sol.-:
a = 7 , b = -48 aur  c = -7
Ab "a×c" = 7×(-7) = -49
Factor of  "a×c" = -49 = 1×(-49)
aur              " b " = -48 = 1-49
7x²-48x-7 = 7x²-(1-49)x-7
                     = 7x²-x+49x-7
                     = (7x²-x)+(49x-7)
                     = x(7x-1)+7(7x-1)
                     = (7x-1)(x+7)     Ans.

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Q-10.  4x²+12x+5 ka factors maalum kijiye.

Sol.-:  
a = 4 , b = 12 aur  c = 5
Ab "a×c" = 4×5 = 20
Factor of  "a×c" = 20 = 10×2
aur              " b " = 12 = 10+2
4x²+12x+5 = 4x²+(10+2)x+5
                     = 4x²+10x+2x+5
                     = (4x²+10x)+(2x+5)
                     = 2x(2x+5)+1(2x+5)
                     = (2x+5)(2x+1)     Ans.



Aaj k liye bas itana hi friends . Mujhe ummid hai ki , ab aap quadratic expression ka factor karna shikh gaye honge.

Also read -:
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• Top 10 math tricks post october 2016

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