Aaj mai factoring of quadratic expression k baare me baat karunga. quadratic exprssion ka factoring , means factor kaise karte hai ? Ya quadratic expression ka gununkhand kaise karte hai? Ye kya hota hai ? inn sabhi k baare me mai full details me aapko bataunga.
Isse hindi me hum "द्विघातीय व्यंजको का गुणनखंड " ( dvighaateey vyanjako ka gunankhand ) khate hai.
General quadratic expression ko hum ax²+bx+c se denote karte hai. Yaha ye yaad rakhe ki a, b aur c numbers (constant) hota hai aur a is not equal to zero hota hai.
Matlab ki "a" ki value "0" ( zero ) nhi ho sakti.
Aaj mai unn quadratic expression ka factor karna bataunga jink liye quadratic expression ax²+bx+c me
a = 1 ho.
a = 2,3,4.... k liye mai next post me bataunga.
Matalab ki aaj mai aapko x²+bx+c type k quadratic expression ka factor karna shikhaunga.
Also read -:
• Top 10 math tricks post november 2016
• Subtraction trick without using borrowing for 2-digit number
Aayiye aaj ka topic start karte hai.
Isse hindi me hum "द्विघातीय व्यंजको का गुणनखंड " ( dvighaateey vyanjako ka gunankhand ) khate hai.
General quadratic expression ko hum ax²+bx+c se denote karte hai. Yaha ye yaad rakhe ki a, b aur c numbers (constant) hota hai aur a is not equal to zero hota hai.
Matlab ki "a" ki value "0" ( zero ) nhi ho sakti.
Aaj mai unn quadratic expression ka factor karna bataunga jink liye quadratic expression ax²+bx+c me
a = 1 ho.
a = 2,3,4.... k liye mai next post me bataunga.
Matalab ki aaj mai aapko x²+bx+c type k quadratic expression ka factor karna shikhaunga.
Also read -:
• Top 10 math tricks post november 2016
• Subtraction trick without using borrowing for 2-digit number
Aayiye aaj ka topic start karte hai.
Factoring of quadratic expressions
Aapko koi problem naa iss liye maine isska working rule niche diya hai . Aap step-by-step rule ko follow kareiye.
Step-1. Number "c" ka factor kariye.
Step-2. "c" k 2 aise factor choose kare . jinko Add ya subtract karene par result "b" k equal ho.
For ex.-: suppose karte "c" k 2 factor s and t hai . Tab c = s×t aur
b = s+t ya b = s - t
Yaha ye yaad rakhe ki, agar quadratic expression me "c" ki value positive (+c) di ho to b= s+t ka use aur negative ( - c) ho to , b= s-t ka use karenge.
Step-3. Ab diye gaye quadratic expression me "b" ki jagah par b = s+t ya b = s - t rakhe.
For ex.-: x²+(s+t)x+c
Step-4. Ab 2-2 ki group banaye . Yahi bana group diye huye quadratic expression ka factors hai.
Also read -:
• Triangle and its types and defination in hindi
• 10 seconds me area of the triangle question solve
Aayiye Ab hum kuch question ko solve karte hai .
Q-1. x²+5x+6 k gununkhand ( factor ) gyat kijiye.
Sol.-: quadratic expression x²+5x+6 me , b = 5 aur c = 6
Factor of "c" = 6 = 2×3
aur " b" = 5 = 2+3
x²+5x+6 = x²+(2+3)x+6
= x²+2x+3x+6
= (x²+2x)+(3x+6)
= x(x+2)+3(x+2)
= (x+2)(x+3) Answer.
Q-2. Find the factor of quadratic expression x²+15x+56
Sol.-: b = 15 aur c = 56
Factors of "c" = 56 = 7×8
aur "b" = 15 = 7+8
x²+15x+56 = x²+(7+8)x+56
= x²+7x+8x+56
= (x²+7x)+(8x+56)
= x(x+7)+8(x+7)
= (x+7)(x+8) Answer.
Q-3. x²+3x-10 k gununkhand gyat kijiye.
Sol.-: b = 3 aur c = -10
Factors of "c" = 10 = 5×2
aur "b" = 3 = 5 - 2
x²+3x-10 = x²+(5-2)x-10
= x²+5x-2x-10
= (x²+5x)-(2x+10)
= x(x+5)-2(x+5)
= (x-2)(x+5) Answer.
Q-4. x²-25x-84 ka factor kariye.
Sol.-: b = -25 aur c = -84
Factors of "c" = 84 = 21×4
aur " b " = 25 = 21+4
x²-25x-84 = x²-(21+4)x-84
= x²-21x-4x-84
= (x²-21x)-(4x+84)
= x(x-21)-4(x-21)
= (x-21)(x-4) Answer.
Q-5. x²+4x-45 ka gununkhand find kijiye.
Sol.-: b = 4 aur c = -45
Factors of "c" = 45 = 9×5
aur " b " = 4 = 9-5
x²+4x-45 = x²+(9-5)x-45
= x²+9x-5x-45
= (x²+9x)-(5x+45)
= x(x+9)-5(x+9)
= (x+9)(x-5) Answer.
Q-6. 20+9x+x² ka factors gyat kijiye.
Sol.-: Sabse pahle isse standard form me likhe -:
x²+9x+20
b = 9 aur c = 20
Factors of "c" = 20 = 5×4
aur " b " = 9 = 5+4
x²+9x+20 = x²+(5+4)x+20
= x²+5x+4x+20
= (x²+5x)+(4x+20)
= x(x+5)+4(x+5)
= (x+5)(x+4) Answer.
Sol.-: sabse pahle isse Standerd form me likhe -:
x²-7x+12
b = -7 aur c = 12
Factors of "c" = 12 = 3×4
aur " b " = 7 = 3+4
x²-7x+12 = x²-(3+4)x+12
= x²-3x-4x+12
= (x²-3x)-(4x-12)
= x(x-3)-4(x-3)
= (x-3)(x-4) Answer.
Also read -:
• square root in few seconds
• Fast calculate square of numbers ending with 5
Q-8. x²+2x-63 ka gununkhand gyat kijiye.
Sol. -: b = 2 aur c = -63
Factors of "c" = 63 = 9×7
aur " b " = 2 = 9-7
x²+2x-63 = x²+(9-7)x-63
= x²+9x-7x-63
= (x²+9x)-(7x+63)
= x(x+9)-7(x+9)
= (x+9)(x-7) Answer.
Q-9. Find the factors of x²-14x-48
Sol.-: b = -14 aur c = -48
Factors of "c" = 48 = 6×8
aur "b" = 14 = 6+8
x²-14x-48 = x²-(6+8)x-48
= x²-6x-8x-48
= (x²-6x)-(8x+48)
= x(x-6)-8(x-6)
= (x-6)(x-8) Answer.
Q-10. x²-14x+49 ka gununkhand kijiye.
Sol.-: b = -14 aur c = 49
Factors of "c" = 49 = 7×7
aur "b" = 14 = 7+7
x²-14x+49 = x²-(7+7)x+49
= x²-7x-7x+49
= (x²-7x) - (7x-49)
= x(x-7)-7(x-7)
= (x-7)(x-7) Answer.
Or
= (x-7)² Answer.
Also read -:
• Fast multiply by 11 in Hindi
Aaj k liye bus itana hi friends, mujhe puri umeed hai ki aap quadratic expression k factors karna shikh gaye honge.
Agar aapko koi problem ho to aap mujhe comment kark puchh sakte hai. Aapki help kark mujhe bahut khushi hogi.Aagar aap humari help karna chhahte hai to bas iss post ko share kare.
Working rules for factoring of quadratic expression
Suppose karte hai ki , hume x²+bx+c ka factor karna hai toStep-1. Number "c" ka factor kariye.
Step-2. "c" k 2 aise factor choose kare . jinko Add ya subtract karene par result "b" k equal ho.
For ex.-: suppose karte "c" k 2 factor s and t hai . Tab c = s×t aur
b = s+t ya b = s - t
Yaha ye yaad rakhe ki, agar quadratic expression me "c" ki value positive (+c) di ho to b= s+t ka use aur negative ( - c) ho to , b= s-t ka use karenge.
Step-3. Ab diye gaye quadratic expression me "b" ki jagah par b = s+t ya b = s - t rakhe.
For ex.-: x²+(s+t)x+c
Step-4. Ab 2-2 ki group banaye . Yahi bana group diye huye quadratic expression ka factors hai.
Also read -:
• Triangle and its types and defination in hindi
• 10 seconds me area of the triangle question solve
Aayiye Ab hum kuch question ko solve karte hai .
Quadratic expression se related question with solution
Q-1. x²+5x+6 k gununkhand ( factor ) gyat kijiye.
Sol.-: quadratic expression x²+5x+6 me , b = 5 aur c = 6
Factor of "c" = 6 = 2×3
aur " b" = 5 = 2+3
x²+5x+6 = x²+(2+3)x+6
= x²+2x+3x+6
= (x²+2x)+(3x+6)
= x(x+2)+3(x+2)
= (x+2)(x+3) Answer.
Q-2. Find the factor of quadratic expression x²+15x+56
Sol.-: b = 15 aur c = 56
Factors of "c" = 56 = 7×8
aur "b" = 15 = 7+8
x²+15x+56 = x²+(7+8)x+56
= x²+7x+8x+56
= (x²+7x)+(8x+56)
= x(x+7)+8(x+7)
= (x+7)(x+8) Answer.
Q-3. x²+3x-10 k gununkhand gyat kijiye.
Sol.-: b = 3 aur c = -10
Factors of "c" = 10 = 5×2
aur "b" = 3 = 5 - 2
x²+3x-10 = x²+(5-2)x-10
= x²+5x-2x-10
= (x²+5x)-(2x+10)
= x(x+5)-2(x+5)
= (x-2)(x+5) Answer.
Sol.-: b = -25 aur c = -84
Factors of "c" = 84 = 21×4
aur " b " = 25 = 21+4
x²-25x-84 = x²-(21+4)x-84
= x²-21x-4x-84
= (x²-21x)-(4x+84)
= x(x-21)-4(x-21)
= (x-21)(x-4) Answer.
Q-5. x²+4x-45 ka gununkhand find kijiye.
Sol.-: b = 4 aur c = -45
Factors of "c" = 45 = 9×5
aur " b " = 4 = 9-5
x²+4x-45 = x²+(9-5)x-45
= x²+9x-5x-45
= (x²+9x)-(5x+45)
= x(x+9)-5(x+9)
= (x+9)(x-5) Answer.
Sol.-: Sabse pahle isse standard form me likhe -:
x²+9x+20
b = 9 aur c = 20
Factors of "c" = 20 = 5×4
aur " b " = 9 = 5+4
x²+9x+20 = x²+(5+4)x+20
= x²+5x+4x+20
= (x²+5x)+(4x+20)
= x(x+5)+4(x+5)
= (x+5)(x+4) Answer.
#Q-7. 12-7x+x² ka factors gyat kijiye.
Sol.-: sabse pahle isse Standerd form me likhe -:
x²-7x+12
b = -7 aur c = 12
Factors of "c" = 12 = 3×4
aur " b " = 7 = 3+4
x²-7x+12 = x²-(3+4)x+12
= x²-3x-4x+12
= (x²-3x)-(4x-12)
= x(x-3)-4(x-3)
= (x-3)(x-4) Answer.
• square root in few seconds
• Fast calculate square of numbers ending with 5
Q-8. x²+2x-63 ka gununkhand gyat kijiye.
Sol. -: b = 2 aur c = -63
Factors of "c" = 63 = 9×7
aur " b " = 2 = 9-7
x²+2x-63 = x²+(9-7)x-63
= x²+9x-7x-63
= (x²+9x)-(7x+63)
= x(x+9)-7(x+9)
= (x+9)(x-7) Answer.
Sol.-: b = -14 aur c = -48
Factors of "c" = 48 = 6×8
aur "b" = 14 = 6+8
x²-14x-48 = x²-(6+8)x-48
= x²-6x-8x-48
= (x²-6x)-(8x+48)
= x(x-6)-8(x-6)
= (x-6)(x-8) Answer.
Q-10. x²-14x+49 ka gununkhand kijiye.
Sol.-: b = -14 aur c = 49
Factors of "c" = 49 = 7×7
aur "b" = 14 = 7+7
x²-14x+49 = x²-(7+7)x+49
= x²-7x-7x+49
= (x²-7x) - (7x-49)
= x(x-7)-7(x-7)
= (x-7)(x-7) Answer.
Or
= (x-7)² Answer.
Also read -:
• Fast multiply by 11 in Hindi
Aaj k liye bus itana hi friends, mujhe puri umeed hai ki aap quadratic expression k factors karna shikh gaye honge.
Agar aapko koi problem ho to aap mujhe comment kark puchh sakte hai. Aapki help kark mujhe bahut khushi hogi.Aagar aap humari help karna chhahte hai to bas iss post ko share kare.
कोई टिप्पणी नहीं:
एक टिप्पणी भेजें