What is Quadratic equation?
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Friends, mera naam hai Dheeraj Sahni aur me apni iss website par aapko explain karta hoon maths se related complex topics in simple and easy way with solved examples !
Friends main aaj aapko bataunga quadratic equation ke baare mein. Aaj is article mein hum sikhenge ki quadratic equation kise kehte hain aur quadratic equations se related kuch historical facts ke baare mein bhi jaanenge. Friends aaj iss article mein me aapko ye bhi sikhaaonga ki agar hame koi equation diya ho to hum kaise check karein ki wo equation quadratic equation hai ya nahi. Aur me aapko iss article me ye bhi bataane wala hoon ki hame quadratic equations ki jaaroorat kyu padti hai matlab hamari daily life me quadratic equation ka kya use hai !
To friends, jyada time waste naa karte hue chailye main topic par aate hain. So, friends...
Friends aaj ka topic hai–Quadratic Equation
Quadratic Equation
Bohot se log aisa maante hain ki wo Babylonians(Babylon ek bohot hi puraani civilization thi aur yaha ke logo lo Babylonians kaha jaata hai) hi the jinhone sabse pehle quadratic equations ko solve karne ka tarikaa pata kiya tha. For instance, wo log ye jaante the ki kaise wo do(two) positive numbers find karte hain agar unn numbers ka positive sum diya hai aur unn numbers ka positive product diya hai. Aur ye problem kisi quadratic equation, jo ki iss form ka hai 👉 x2–px+q=0, ko solve karne ke equivalent(equal) hai. Greek mathematician Euclid ne ek geometrical concept/approach develop kiya jiske use se lengths pata ki jaati thi. Iss cheez ko hum aaj quadratic equation ke solution kehte hain. Kisi quadratic equation, jo ki apne general form me ho, ko solve karne ka credit praay(often) ancient Indian mathematician ko diya jata hai. In fact Brahmagupta ne ek spashta(explicit) formula diya tha kisi quadratic equation, jiska form ax2 + bx = c ho, ko solve karne ka. Baad mein Sridharacharya(A.D. 1025) ne ek formula nikala (derive kiya) jise ab hum quadratic formula kehte hain (as quoted by Bhaskara II). Iss formula ka use kisi quadratic equation ko completing the square method se solve karne ke liye kiya jata hai. Iss formula ke use se hum kisi bhi quadratic equation ko bahut aasani se (easily) aur kam time mein solve kar sakte hain. Ek Arab mathematician Al-Khwarizmi (about A.D. 800) ne bhi different types ke quadratic equation ki study ki thi. Abraham bar Hiyya Ha-Nasi ne apni ek book ‘ Liber embadorum ’ Europe main A.D. 1145 mein published ki thi. Uss book mein different types ke quadratic equations ke complete solutions hain.
To chaliye ab jante hain ki quadratic equation kise kehte hain.
Ek quadratic equation jisme variable x hai aisa equation hota hai jiska roop/form ax2 + bx + c = 0 ho jahan par a,b aur c real numbers hain aur a≠0( b aur c, 0 ho sakte hain). Example ke liye: 3x2 – 2x + 1 = 0 ye ek quadratic equation kyuki 3x2 – 2x + 1 = 0 ko ax2 + bx + c = 0 ke form mein likha ja sakta hai [3x2 – 2x + 1 = 0 ⇒ 3x2 + (–2)x + 1 = 0] jahan par a = 3, b = –2, c = 1 hai.
[Agar aapko nahi pata ki real number kise kehte hain to ye jaanne ke liye iss blue link par click kijiye»»
What is complex number,real number,irrational number, rational number,integer,whole number,natural number?].
Isi tarah se x2 – 1 = 0 quadratic equation hai Kyunki x2 – 1 = 0 ko ax2 + bx + c = 0 ke form me likha jaa sakta hai [x2 – 1 = 0 ⇒ x2 + (0)x – 1 = 0 ⇒ (1)x2 + (0)x + (–1) = 0] jahan par a = 1, b = 0, c = –1 hai.
Isi tarah se x – x2 = 0 bhi ek quadratic equation hai kyunki x – x2 = 0 ko ax2 + bx + c = 0 ke form mein likha ja sakta hai [x – x2 = 0 ⇒ –x2 + x = 0 ⇒ –x2 + x + 0 = 0 ⇒ (–1)x2 + (1)x + 0 = 0] jahan par a = –1, b = 1, c = 0 hai.
In fact, koi bhi equation jo ki p(x) = 0 ke form mein ho aur p(x) ek aisa polynomial ho jiski degree 2 ho, ek quadratic equation kehlata hai. Lekin jab hum polynomial p(x) ke terms jo unke degrees ke descending order mein likhte hain toh hame us a equation ka standard form milta hai. Kehne ka matlab bas yeh hai ki ax2 + bx + c = 0, a≠0 ko quadratic equation ka standard form kehte hain.
To ab aap samajh hi gaye honge ki quadratic equation kya hota hai.
[Agar aap polynomial, equation, term, degree etc. ke baare mein nahi jante hain to ye jaanne ke liye iss blue link par click kijiye. Isko padhne ke baad aap aur sahi se quadratic equation ko samajh sakte hain »»»
What is polynomial and equation?].
Hamari daily life situations mein quadratic equation aate hain. For instance,
(a) Suppose a charity trust decides to build a prayer hall having a carpet area of 300 square metres with its length one metre more than twice its breadth. What should be the length and breadth of the hall?
[Maan lijiye ek charity trust ek prayer hall banana chahti hai jiska carpet area 300 square metre jiski length uski breadth ke dugune (double/twice) se ek(one) metre jyada ho. To uss hall ki length aur breadth kya honi chahiye?]
Solution: Maan lijiye uss hall ki breath x metres hai to uss hall ki length (2x+1) metres hui (according to question). Hum iss information ko pictorially aise dikha sakte hain (neeche diya gaya yeh diagram dekhiye).
Now area of the hall = (2x+1).x m2
So, 2x2 + x = 300 (Given)
Therefore, 2x2 + x – 300 = 0
So, hall ki breadth equation 2x2 + x – 300 = 0 ko satisfy karna chahiye jo ki ek quadratic equation hai.
(b) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with.
[John aur Jivanti dono ke paas total 45 marbles hain. Unme se dono ne apne 5-5 marbles kho diya aur ab jo unke paas marbles bach gaye unka product 124 hai. To hame batana hai ki unke paas starting mein kitne marbles the?]
Solution: Let the number of Marbles John had be x.
Then the number of marbles Jivanti had = 45–x.
The number of marbles left with John when he lost 5 marbles = x–5.
The number of marbles left with Jivanti when she lost 5 marbles = 45–x–5
= 40–x
So, their product= (x–5)(40–x)
= 40x–x2–200+5x
= –x2+45x–200
So, –x2+45x–200=124 (product =124)
–x2+45x–324=0
x2–45x+324=0
So, John ke paas jitne marbles hain unhe iss equation x2–45x+324=0 ko satisfy karna chahiye jo ki ek quadratic equation hai.
(c) A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day the total cost of production was Rs 750. We would like to find out the number of toys produced on that day.
[Ek cottage industry ek din mein 6 toys banati hai har(each) toy ki keemat(cost), rupees mein, hai 55 minus ek din mein jitne toys bante hain(produce hote hain) Kisi vishesh(particular) din(day) par jitne toys bane(produce huye) unki total cost Rs. 750 thi. To hame batana hai ki uss din kitne toys bane the (produce hue the]
Solution: Let the number of toys produce on that day be x.
Therefore, the cost of production (in rupees) of each toy that day = 55 – x.
So, the total cost of production (in rupees) that day = x(55–x).
Therefore, x(55–x) = 750
55x–x2 = 750
–x2+55x–750=0
x2–55x+750=0
So, uss din jitne toys produce hue the ko iss equation x2–55x+750=0 ko satisfy karna chahiye jo ki ek quadratic equation hai.
To abhi aapne dekha ki kis tarah hamare daily life mein quadratic equation ka use hota hai.
Chaliye ab aage badhate hain. Maan Lijiye aap ko ek equation diya hai jo ki dekhne se quadratic equation nahi lag raha hai to aap kaise pata karenge ki wo equation quadratic equation hai ya phir kisi aur type ka equation hai. Chaliye kuch examples ke through ise samajhte hain...
Check whether the following are quadratic equations:
[Check kijiye kya diya gye equations quadratic equations hain]
(a) (x-2)2+1=2x-3
Solution:
(b) (x+1)2=2(x–3)
Solution:
(c) (x+2)3=2x(x2–1)
Solution:
So friends, iss article main apne seekha ki quadratic equation kya hota hai. Apne next article mein me aap ko bataunga ki kisi quadratic equation ko kaise solve karte hain ?
I hope ki aapko mera ye article pasand aaya hoga. I hope ki aapko sab kuch acche se samajh me aa gaya hoga. Aur agar aapko koi cheez samajh me nahi aayi ho to commemt ke through aap hame bata sakte hain. Agar aapko ye post accha laga ho to comment ke maadhyam se aap hame support kar sakte hai.
Aapne is article ko padhkar kya seekha?
CHECK YOUR KNOWLEDGE
Give answers:
Q.1-The area of a rectangular plot is 528 m2. The length of the plot (in metres) is 1 more than twice its breadth. Express this situation in the form of quadratic equation.
Q.2- Is 2x–3=(x+4)3 a quadratic equation.
Q.3-Express y2=5 in standard form.
Aap apne answers hame comment ke through bata sakte hain.
THANKS FOR READING THIS BLOG.
Friends, mera naam hai Dheeraj Sahni aur me apni iss website par aapko explain karta hoon maths se related complex topics in simple and easy way with solved examples !
Friends main aaj aapko bataunga quadratic equation ke baare mein. Aaj is article mein hum sikhenge ki quadratic equation kise kehte hain aur quadratic equations se related kuch historical facts ke baare mein bhi jaanenge. Friends aaj iss article mein me aapko ye bhi sikhaaonga ki agar hame koi equation diya ho to hum kaise check karein ki wo equation quadratic equation hai ya nahi. Aur me aapko iss article me ye bhi bataane wala hoon ki hame quadratic equations ki jaaroorat kyu padti hai matlab hamari daily life me quadratic equation ka kya use hai !
To friends, jyada time waste naa karte hue chailye main topic par aate hain. So, friends...
Let's begin...
Friends aaj ka topic hai–Quadratic Equation
Quadratic Equation
Bohot se log aisa maante hain ki wo Babylonians(Babylon ek bohot hi puraani civilization thi aur yaha ke logo lo Babylonians kaha jaata hai) hi the jinhone sabse pehle quadratic equations ko solve karne ka tarikaa pata kiya tha. For instance, wo log ye jaante the ki kaise wo do(two) positive numbers find karte hain agar unn numbers ka positive sum diya hai aur unn numbers ka positive product diya hai. Aur ye problem kisi quadratic equation, jo ki iss form ka hai 👉 x2–px+q=0, ko solve karne ke equivalent(equal) hai. Greek mathematician Euclid ne ek geometrical concept/approach develop kiya jiske use se lengths pata ki jaati thi. Iss cheez ko hum aaj quadratic equation ke solution kehte hain. Kisi quadratic equation, jo ki apne general form me ho, ko solve karne ka credit praay(often) ancient Indian mathematician ko diya jata hai. In fact Brahmagupta ne ek spashta(explicit) formula diya tha kisi quadratic equation, jiska form ax2 + bx = c ho, ko solve karne ka. Baad mein Sridharacharya(A.D. 1025) ne ek formula nikala (derive kiya) jise ab hum quadratic formula kehte hain (as quoted by Bhaskara II). Iss formula ka use kisi quadratic equation ko completing the square method se solve karne ke liye kiya jata hai. Iss formula ke use se hum kisi bhi quadratic equation ko bahut aasani se (easily) aur kam time mein solve kar sakte hain. Ek Arab mathematician Al-Khwarizmi (about A.D. 800) ne bhi different types ke quadratic equation ki study ki thi. Abraham bar Hiyya Ha-Nasi ne apni ek book ‘ Liber embadorum ’ Europe main A.D. 1145 mein published ki thi. Uss book mein different types ke quadratic equations ke complete solutions hain.
To chaliye ab jante hain ki quadratic equation kise kehte hain.
Ek quadratic equation jisme variable x hai aisa equation hota hai jiska roop/form ax2 + bx + c = 0 ho jahan par a,b aur c real numbers hain aur a≠0( b aur c, 0 ho sakte hain). Example ke liye: 3x2 – 2x + 1 = 0 ye ek quadratic equation kyuki 3x2 – 2x + 1 = 0 ko ax2 + bx + c = 0 ke form mein likha ja sakta hai [3x2 – 2x + 1 = 0 ⇒ 3x2 + (–2)x + 1 = 0] jahan par a = 3, b = –2, c = 1 hai.
[Agar aapko nahi pata ki real number kise kehte hain to ye jaanne ke liye iss blue link par click kijiye»»
What is complex number,real number,irrational number, rational number,integer,whole number,natural number?].
Isi tarah se x – x2 = 0 bhi ek quadratic equation hai kyunki x – x2 = 0 ko ax2 + bx + c = 0 ke form mein likha ja sakta hai [x – x2 = 0 ⇒ –x2 + x = 0 ⇒ –x2 + x + 0 = 0 ⇒ (–1)x2 + (1)x + 0 = 0] jahan par a = –1, b = 1, c = 0 hai.
In fact, koi bhi equation jo ki p(x) = 0 ke form mein ho aur p(x) ek aisa polynomial ho jiski degree 2 ho, ek quadratic equation kehlata hai. Lekin jab hum polynomial p(x) ke terms jo unke degrees ke descending order mein likhte hain toh hame us a equation ka standard form milta hai. Kehne ka matlab bas yeh hai ki ax2 + bx + c = 0, a≠0 ko quadratic equation ka standard form kehte hain.
To ab aap samajh hi gaye honge ki quadratic equation kya hota hai.
[Agar aap polynomial, equation, term, degree etc. ke baare mein nahi jante hain to ye jaanne ke liye iss blue link par click kijiye. Isko padhne ke baad aap aur sahi se quadratic equation ko samajh sakte hain »»»
What is polynomial and equation?].
Hamari daily life situations mein quadratic equation aate hain. For instance,
(a) Suppose a charity trust decides to build a prayer hall having a carpet area of 300 square metres with its length one metre more than twice its breadth. What should be the length and breadth of the hall?
[Maan lijiye ek charity trust ek prayer hall banana chahti hai jiska carpet area 300 square metre jiski length uski breadth ke dugune (double/twice) se ek(one) metre jyada ho. To uss hall ki length aur breadth kya honi chahiye?]
Solution: Maan lijiye uss hall ki breath x metres hai to uss hall ki length (2x+1) metres hui (according to question). Hum iss information ko pictorially aise dikha sakte hain (neeche diya gaya yeh diagram dekhiye).
Now area of the hall = (2x+1).x m2
So, 2x2 + x = 300 (Given)
Therefore, 2x2 + x – 300 = 0
So, hall ki breadth equation 2x2 + x – 300 = 0 ko satisfy karna chahiye jo ki ek quadratic equation hai.
(b) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with.
[John aur Jivanti dono ke paas total 45 marbles hain. Unme se dono ne apne 5-5 marbles kho diya aur ab jo unke paas marbles bach gaye unka product 124 hai. To hame batana hai ki unke paas starting mein kitne marbles the?]
Solution: Let the number of Marbles John had be x.
Then the number of marbles Jivanti had = 45–x.
The number of marbles left with John when he lost 5 marbles = x–5.
The number of marbles left with Jivanti when she lost 5 marbles = 45–x–5
= 40–x
So, their product= (x–5)(40–x)
= 40x–x2–200+5x
= –x2+45x–200
So, –x2+45x–200=124 (product =124)
–x2+45x–324=0
x2–45x+324=0
So, John ke paas jitne marbles hain unhe iss equation x2–45x+324=0 ko satisfy karna chahiye jo ki ek quadratic equation hai.
(c) A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day the total cost of production was Rs 750. We would like to find out the number of toys produced on that day.
[Ek cottage industry ek din mein 6 toys banati hai har(each) toy ki keemat(cost), rupees mein, hai 55 minus ek din mein jitne toys bante hain(produce hote hain) Kisi vishesh(particular) din(day) par jitne toys bane(produce huye) unki total cost Rs. 750 thi. To hame batana hai ki uss din kitne toys bane the (produce hue the]
Solution: Let the number of toys produce on that day be x.
Therefore, the cost of production (in rupees) of each toy that day = 55 – x.
So, the total cost of production (in rupees) that day = x(55–x).
Therefore, x(55–x) = 750
55x–x2 = 750
–x2+55x–750=0
x2–55x+750=0
So, uss din jitne toys produce hue the ko iss equation x2–55x+750=0 ko satisfy karna chahiye jo ki ek quadratic equation hai.
To abhi aapne dekha ki kis tarah hamare daily life mein quadratic equation ka use hota hai.
Chaliye ab aage badhate hain. Maan Lijiye aap ko ek equation diya hai jo ki dekhne se quadratic equation nahi lag raha hai to aap kaise pata karenge ki wo equation quadratic equation hai ya phir kisi aur type ka equation hai. Chaliye kuch examples ke through ise samajhte hain...
Check whether the following are quadratic equations:
[Check kijiye kya diya gye equations quadratic equations hain]
(a) (x-2)2+1=2x-3
Solution:
(b) (x+1)2=2(x–3)
Solution:
(c) (x+2)3=2x(x2–1)
Solution:
So friends, iss article main apne seekha ki quadratic equation kya hota hai. Apne next article mein me aap ko bataunga ki kisi quadratic equation ko kaise solve karte hain ?
I hope ki aapko mera ye article pasand aaya hoga. I hope ki aapko sab kuch acche se samajh me aa gaya hoga. Aur agar aapko koi cheez samajh me nahi aayi ho to commemt ke through aap hame bata sakte hain. Agar aapko ye post accha laga ho to comment ke maadhyam se aap hame support kar sakte hai.
Aapne is article ko padhkar kya seekha?
CHECK YOUR KNOWLEDGE
Give answers:
Q.1-The area of a rectangular plot is 528 m2. The length of the plot (in metres) is 1 more than twice its breadth. Express this situation in the form of quadratic equation.
Q.2- Is 2x–3=(x+4)3 a quadratic equation.
Q.3-Express y2=5 in standard form.
Aap apne answers hame comment ke through bata sakte hain.
THANKS FOR READING THIS BLOG.
Comments
Nice article keep it up bro