How to Solve Quadratic Equations using Algebraic Formulae?
Friends maine aapko apne previous article mein quadratic equation ke baare me bataya tha. Aaj me aapko iss article mein bataoonga ki quadratic equation ko kaise solve karte hain. Aur iss article ke last mein me aapko kuch questions doonga. Aapko unn questions ke answers dene honge. Aap unn questions ke answers tab hi de paayenge jab aap is article ko poora padenge. So read this article, increase your knowledge and give answers.
Friends mera naam hai Dheeraj Sahni aur aap log iss waqt hain meri website www.mathshindi.com par.
Friends main aap ko apni website par maths related different complex topics/ concepts ko simple aur easy way mein samjhane ki koshish karta hoon. Friends isi tarah ka ek topic aaj main laya hoon aapke liye. Friends aaj ka hamara topic hai – How to Solve Quadratic Equation Using Algebraic Formulae.
So friends...
Friends hamare daily life mein bahut sari aisi situations aati hai jahan par hum uss situation ko quadratic equation ke form me show kar sakte hain. Example ke liye zara is situation ko dekhiye –
The product of Sunita’s age (in years) 2 years ago and her age 4 years from now is 1 more than twice her present age. What is her present age?
[Do(two) saal pehle Sunita ki jo age thi aur ab se char(four) saal baad jo sunita ki age hogi, unn dono ka product Sunita ki jo ab age hai (yani ki uski present age) ke dogune(double/twice) se ek(one) jyada hai].
Iss situation/problem ko mathematically kaise show karein, aaiye dekhte hain–
Let Sunita's present age = x years
Two years ago her age was = (x–2) years Four years from now her age will be = (x+4) years
So according to question,
(x – 2)(x + 4) = 2x + 1
x(x + 4) – 2(x + 4) = 2x + 1
x2 + 4x – 2x – 8 = 2x + 1
x2 + 2x – 8 = 2x + 1
x2 + 2x – 8 – 2x – 1 = 0
x2 + 2x – 2x – 8 – 1 = 0
x2 – 9 = 0
Ye equation x2 – 9 = 0 ek quadratic equation hai. Ye equation x2 – 9 = 0 iss situation ko mathematically show karta hai.To agar hum kisi tarah se is equation x2 – 9 = 0 solve kar de to hum iss situation/problem ko bhi solve kar paayenge.
[What is Quadratic Equation ? – iss post ko padhe ke liye iss blue link par click kijiye »»»
What is quadratic equation ?].
To chaliye ab me aapko batata hoon ki kisi quadratic equation ko kaise solve karte hain. Kisi quadratic equation ko solve karne ke chaar(four) tarike methods hote hain–
1. Using algebraic formulae
2. Using middle term splitting method
3. Completing square method
4. Using quadratic formula
Yaha par me inn charo mein se sirf ek hi tarika/method ke baare mein bataoonga– Using algebraic formulae.
To kisi quadratic equation ko algebraic formulae se solve karne se pehle aap ko yeh cheezein pata honi chahiye:
(a) Meaning of solution of a equation: Kisi equation ko solve karne ka matlab hota hai ki aise number(s) find karna jinhe uss equation mein variable ki jagah/place par rakhne par uss equation ke LHS aur RHS equal ho jaayein.
Example ke liye iss equation ko lijiye: x2 + 2x – 143 = 0
Iss quadratic equation main x ki jagah 11 rakhne par (putting x = 11),
LHS = (11)2 + 2(11) – 143
= 121 + 22 – 143
= 143 – 143
= 0 = RHS
LHS, RHS ke barabar (equal) aaya hai. So, 11 iss quadratic equation x2 + 2x – 143 = 0 ka ek solution hai.
Isi tarah se iss quadratic equation main x ki jagah –13 rakhne par (putting x = –13),
LHS = (–13)2 + 2(–13) – 143
= 169 – 26 – 143
= 169 – 169
= 0 = RHS
LHS, RHS ke barabar (equal) aaya hai. So, –13 iss quadratic equation x2 + 2x – 143 = 0 ka ek solution hai.
(b) “Fundamental theorem of algebra: Any polynomial has as many solution as the degree of the polynomial.”
Matlab kisi polynomial ke utne hi solution(s) hote hain jitni uss polynomial ki degree hoti hai. Yeh baat kisi equation ke liye bhi true hai. “Kisi bhi equation ke utne hi solution(s) hote hai jitni uski degree hoti hai.” Jaisa ki abhi aapne dekha ki quadratic equation x2 + 2x – 143 = 0 ke do hi solutions: 11 aur –13 hai kyunki yeh ek quadratic equation hai matlab kyuki iss equation x2 + 2x – 143 = 0 ki degree 2 hai.
[Agar aap polynomial, term, degree etc. ke baare mein nahi jante hain to ye jaanne ke liye iss blue link per click kijiye. Isko padhne ke baad aap aur sahi se quadratic equation ko samajh sakte hain »»»
What is polynomial and equation?].
(c) Algebraic formulae:
(i) (a + b)2 = a2 + 2ab + b2
(ii) (a – b)2 = a2 – 2ab + b2
(iii) a2 – b2 = (a + b)(a – b)
Ye teen(three) common algebraic formulae hai jo Maths mein har jagah use hote hain.
[Note:- Solutions, zeroes aur roots– inn teeno ka matlab ek hi hota hai. Ye teeno synonym words hai].
To chaliye ab hum kuch quadratic equation ko algebraic formulae ke use se solve karte hain...
How to Solve Quadratic Equations using Algebraic Formulae ?
Ex.1. Find the roots of quadratic equation x2 + 6x + 9 = 0 using algebraic formula.
Solution:
Ex.2. Find the roots of quadratic equation x2 + 12x + 36 = 0 using algebraic formula.
Solution:
Ex.3. Find the roots of quadratic equation 4x2 – 36x + 81 = 0 using algebraic formula.
Solution:
Ex.4. Find the roots of quadratic equation 8x2 – 40x + 50 = 0 using algebraic formula.
Solution:
Ex.5. Find the roots of quadratic equation x2 – 4 = 0 using algebraic formula.
Solution:
Ex.6. Find the roots of quadratic equation 3x2 – 27 = 0 using algebraic formula.
Solution:
To iss article mein maine aapko sirf algebraic formula ka use karke kisi quadratic equation ko kaise solve karte hain sirf issi ke baare me bataya hai aur baaki ke methods ke baare mein me apne next article main aapko bataoonga.
Agar aapko a article pasand aaya to comment ke madhyam se hame bata sakte hain aur hame support kar sakte hain. Mujhe umeed hai ki aap ke liye ye article helpful prove hua hoga.
Aapne is article ko padhkar kya seekha?
CHECK YOUR KNOWLEDGE
Give answers:
Q.1- Find the roots of quadratic equation 25x2 + 90x + 9 = 0 using algebraic formula.
Q.2- Find the roots of quadratic equation 49x2 – 56x + 4 = 0 using algebraic formula.
Q.3- Find the roots of quadratic equation x2 – 36 = 0 using algebraic formula.
Aap apne answers hame comment ke through bata sakte hain.
THANKS FOR READING THIS BLOG.
Friends mera naam hai Dheeraj Sahni aur aap log iss waqt hain meri website www.mathshindi.com par.
Friends main aap ko apni website par maths related different complex topics/ concepts ko simple aur easy way mein samjhane ki koshish karta hoon. Friends isi tarah ka ek topic aaj main laya hoon aapke liye. Friends aaj ka hamara topic hai – How to Solve Quadratic Equation Using Algebraic Formulae.
So friends...
Let's begin...
Friends hamare daily life mein bahut sari aisi situations aati hai jahan par hum uss situation ko quadratic equation ke form me show kar sakte hain. Example ke liye zara is situation ko dekhiye –
The product of Sunita’s age (in years) 2 years ago and her age 4 years from now is 1 more than twice her present age. What is her present age?
[Do(two) saal pehle Sunita ki jo age thi aur ab se char(four) saal baad jo sunita ki age hogi, unn dono ka product Sunita ki jo ab age hai (yani ki uski present age) ke dogune(double/twice) se ek(one) jyada hai].
Iss situation/problem ko mathematically kaise show karein, aaiye dekhte hain–
Let Sunita's present age = x years
Two years ago her age was = (x–2) years Four years from now her age will be = (x+4) years
So according to question,
(x – 2)(x + 4) = 2x + 1
x(x + 4) – 2(x + 4) = 2x + 1
x2 + 4x – 2x – 8 = 2x + 1
x2 + 2x – 8 = 2x + 1
x2 + 2x – 8 – 2x – 1 = 0
x2 + 2x – 2x – 8 – 1 = 0
x2 – 9 = 0
Ye equation x2 – 9 = 0 ek quadratic equation hai. Ye equation x2 – 9 = 0 iss situation ko mathematically show karta hai.To agar hum kisi tarah se is equation x2 – 9 = 0 solve kar de to hum iss situation/problem ko bhi solve kar paayenge.
[What is Quadratic Equation ? – iss post ko padhe ke liye iss blue link par click kijiye »»»
What is quadratic equation ?].
To chaliye ab me aapko batata hoon ki kisi quadratic equation ko kaise solve karte hain. Kisi quadratic equation ko solve karne ke chaar(four) tarike methods hote hain–
1. Using algebraic formulae
2. Using middle term splitting method
3. Completing square method
4. Using quadratic formula
Yaha par me inn charo mein se sirf ek hi tarika/method ke baare mein bataoonga– Using algebraic formulae.
To kisi quadratic equation ko algebraic formulae se solve karne se pehle aap ko yeh cheezein pata honi chahiye:
(a) Meaning of solution of a equation: Kisi equation ko solve karne ka matlab hota hai ki aise number(s) find karna jinhe uss equation mein variable ki jagah/place par rakhne par uss equation ke LHS aur RHS equal ho jaayein.
Example ke liye iss equation ko lijiye: x2 + 2x – 143 = 0
Iss quadratic equation main x ki jagah 11 rakhne par (putting x = 11),
LHS = (11)2 + 2(11) – 143
= 121 + 22 – 143
= 143 – 143
= 0 = RHS
LHS, RHS ke barabar (equal) aaya hai. So, 11 iss quadratic equation x2 + 2x – 143 = 0 ka ek solution hai.
Isi tarah se iss quadratic equation main x ki jagah –13 rakhne par (putting x = –13),
LHS = (–13)2 + 2(–13) – 143
= 169 – 26 – 143
= 169 – 169
= 0 = RHS
LHS, RHS ke barabar (equal) aaya hai. So, –13 iss quadratic equation x2 + 2x – 143 = 0 ka ek solution hai.
(b) “Fundamental theorem of algebra: Any polynomial has as many solution as the degree of the polynomial.”
Matlab kisi polynomial ke utne hi solution(s) hote hain jitni uss polynomial ki degree hoti hai. Yeh baat kisi equation ke liye bhi true hai. “Kisi bhi equation ke utne hi solution(s) hote hai jitni uski degree hoti hai.” Jaisa ki abhi aapne dekha ki quadratic equation x2 + 2x – 143 = 0 ke do hi solutions: 11 aur –13 hai kyunki yeh ek quadratic equation hai matlab kyuki iss equation x2 + 2x – 143 = 0 ki degree 2 hai.
[Agar aap polynomial, term, degree etc. ke baare mein nahi jante hain to ye jaanne ke liye iss blue link per click kijiye. Isko padhne ke baad aap aur sahi se quadratic equation ko samajh sakte hain »»»
What is polynomial and equation?].
(c) Algebraic formulae:
(i) (a + b)2 = a2 + 2ab + b2
(ii) (a – b)2 = a2 – 2ab + b2
(iii) a2 – b2 = (a + b)(a – b)
Ye teen(three) common algebraic formulae hai jo Maths mein har jagah use hote hain.
[Note:- Solutions, zeroes aur roots– inn teeno ka matlab ek hi hota hai. Ye teeno synonym words hai].
To chaliye ab hum kuch quadratic equation ko algebraic formulae ke use se solve karte hain...
How to Solve Quadratic Equations using Algebraic Formulae ?
Ex.1. Find the roots of quadratic equation x2 + 6x + 9 = 0 using algebraic formula.
Solution:
Ex.2. Find the roots of quadratic equation x2 + 12x + 36 = 0 using algebraic formula.
Solution:
Solution:
Ex.4. Find the roots of quadratic equation 8x2 – 40x + 50 = 0 using algebraic formula.
Solution:
Solution:
Ex.6. Find the roots of quadratic equation 3x2 – 27 = 0 using algebraic formula.
Solution:
To iss article mein maine aapko sirf algebraic formula ka use karke kisi quadratic equation ko kaise solve karte hain sirf issi ke baare me bataya hai aur baaki ke methods ke baare mein me apne next article main aapko bataoonga.
Agar aapko a article pasand aaya to comment ke madhyam se hame bata sakte hain aur hame support kar sakte hain. Mujhe umeed hai ki aap ke liye ye article helpful prove hua hoga.
Aapne is article ko padhkar kya seekha?
CHECK YOUR KNOWLEDGE
Give answers:
Q.1- Find the roots of quadratic equation 25x2 + 90x + 9 = 0 using algebraic formula.
Q.2- Find the roots of quadratic equation 49x2 – 56x + 4 = 0 using algebraic formula.
Q.3- Find the roots of quadratic equation x2 – 36 = 0 using algebraic formula.
Aap apne answers hame comment ke through bata sakte hain.
THANKS FOR READING THIS BLOG.
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