factoring of quadratic exprssions - 1 me maine aapko x²+bx+c type k question ka factor karna shikhaya tha.
Aaj k liye bas itana hi friends . Mujhe ummid hai ki , ab aap quadratic expression ka factor karna shikh gaye honge.
Also read -:
• Kya aapko koi help mathematics me chahiye
• Top 10 math tricks post october 2016
Agar aapko koi problem ho to aap comment kark puchh sakte hai. Mujhe aapki help kark bahot khushi hogi. Agar aap meri help karna chhahate hai to bas iss topic ko share kare.
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.
Also read -:
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.
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.
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.
Also read -:
• Trigonometry ka basic concept in hindi part-1
• Fast cube tricks 1se 99 tak in Hindi
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.
Also read -:
• Trigonometry ka basic concept in hindi part-1
• Fast cube tricks 1se 99 tak in Hindi
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.
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.
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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• Fast multiply by 11 in Hindi
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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.
Also read -:
• Fast multiply by 11 in Hindi
• Fast multiply by 99,999,9999 of any numbers in hindi
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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• Fast calculate square of numbers ending with 5
• square root in few seconds
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.
Also read -:
• Fast calculate square of numbers ending with 5
• square root in few seconds
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.
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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• Iss fast trick se kariye simple equation instantly solve
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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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• Iss fast trick se kariye simple equation instantly solve
• Mathematics operation
Q-10. 4x²+12x+5 ka factors maalum kijiye.
Sol.-:
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.
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 -:
• Kya aapko koi help mathematics me chahiye
• Top 10 math tricks post october 2016
Agar aapko koi problem ho to aap comment kark puchh sakte hai. Mujhe aapki help kark bahot khushi hogi. Agar aap meri help karna chhahate hai to bas iss topic ko share kare.
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