QUADRATIC EQUATIONS - ONE SHOT | SHORTCUTS + PYQS | HIGH WEIGHTAGE CHAPTER | JEE 2024 | KIRAN SIR

hi hello Namaste W welcome jna welcome welcome welcome important and high weight chapter that is quadratic equations so quadratic equations important so number of questions so Maxim Maxim complete able to solve the questions not okay I'm very good very good very evening good evening to all so classes DEC but it is a sucessful so like for examp just for the de December 21st class start okay right guys so quity okay guys so December okay that is very very useful for you guys use okay sucess series Max okay Avengers Avers okay guys quad equ first of all transformations s transformation trans straight L translation trans Transformations important first one third one and six one points important okay right let us move quadratic equation so first of all qu equ quadratic expression Quadra expression Quadra equation x² - 5x + 6 solab what is the range of this qu x - 5 + 6 = equ so okay expr equ a² BX plus Cal is called quadratic equation this is called the linear equation not the quadratic find the roots of the qura equation root Roots equation Ro for example x² 2x into X = = Ro two Roots how many roots are there there are two Roots you should only one solution only one soltion two Ro Sol only one solution four five first root second root third root one of soltions one is the solution another one is the solution there are two solution 2 2 3 Ro 2 3 4 5 six there are six Roots how many solutions are there Babu S one is a solution two is a solution three is a solution so there are only three soltion only one solution Solutions but Ro six Roots so that is the main difference between roots and soltion okay right bet for a Quadra equation if Al beta The Roots then sum of the roots isus B by a product of the roots is C by first of all good morning good evening very good evening right so Al Bet of the roots by of the roots C right there are so many ways to find out the roots number one a b Cal Roots x = minus b + orus sare root of B s- 4 by 2 x² + x - 6al example a B+ C x² + B by a into x + C by AAL 0 Al Bet root x² x² minus of alpha + beta into x + Alpha Beta is equal to second x squ a and last constant c a value ausus right x + x - 6 = a into c a - 6 - 6 into 1 - 6 2 into -3 - 6 this is the wrong so what is the correct pairus 2 and three - 2 + 3 x² - 2 + 3 into x - 6 = + x² - 2 into x - 2 x + 3 into x 3X - 6 equal 0 from these two x common x - 2 from these two three common x - 2 is equal to 0 from these two take x - - 2 Common then left with x + 3 is equal to 0 so either this 0 or this 0 if this 0 x = 2 if this 0 xal minus 3 so what are the roots sir 2 andus 3 are the roots Roots 2 andus 1 BX B Val 1 C valus 6 x = - b + or - root of b² - 4 into uh a into c c and - 6 by 2 a 2 a and 1 so -1 plus orus sare root of 24 + 24 + 1 25 by 2 -1 + or - 5 by2 so-1 - 5 by2 plusus -1 + 5 by 2- 6 2- 3 4 by 2 are two right Ro of the root Ro b + b c AAL x² minus of - B by a into x + C by AAL 0 Al Bet Ro X into X bet Al into X Plus Alpha betaal 0 comp x xare 1 so xus of alha plus beta andus B by a Alpha plus beta minus B by a and product of roots Alpha Beta bet equation where beta are the roots for this okay right sir so first question number one find the roots of find the roots of roots of x² - 4x + 3 = 0 question process only X = Min - b + orus root of b² - 4 a c by 2 a so x = 4 + r - < TK of 16 - 12 is 4 by 2 so x = 4 + r - 2 by 2 4 + 2 6 by 2 3 2 by 2 1 x = 1A 3 are the roots 4 + 3 - x - 3x + 3al 0 x -- 3 Comm x -1 equal 0 so x- X - 3 is equal 0 so xal either 1 or 3 so 1 comma 3 are the roots for the quadratic clear there is equ for this quadratic for quadratic roots are are Al Bet Ro bet then find the root find the equation for which for which the roots are for which the roots are number one 1 by 1 bet Ro by bet 1 by Al 1 by bet bet 1 by 1 by Beta Ro root root bet Al Square beta square roots equation fourth one 2 by Beta beta alus 2A beta + 2 by betus 2ot one Alp by betaa Beta by Al root different different Ty of question so first second so there are two Roots first root is in terms of Alpha Beta first Ro second plus BX plus c equal Z ala bet Roots so so therefore a Al squ B Alpha Plus [Music] cal0 1 by 1 by Beta root I'm taking xal 1 by alha x value 1 by b 1 square + B into 1 by x + C = 0 a by x² + B by X+ cal0 LC a + b x + C x² 1 by by root alha root bet root root Al sorry let xal root alha X + B+ C x² square B x² plus C = 0 a x 4 + B x² + Cal 0 bet sare so let us take xal Al Square so alare square root on both sides sare root sare root so Alpha equal root X so Alpha equal plus orus rootx so a alpha s + B Alpha + cal0 Alpha plus r- rootx a into plus r- rootx s+ b into plus r- rootx + cal0 a plusus no problem so but we got here two equations 2us 2 Beta + 2 bet so let xal X = Alp + 2 by alpus 2 alal cross x - 2x = Alp + 2 Al Al x 2 2 + 2 Comm x + so this is Alpha = 2 * of x + 1 by X - 1 Cal so a into Al Square 2 S 4 into x + 1 by x -1 s + B into 2 * of x + 1 by x -1 + C = 0 4 a into x +1 s + 2 B into x + 1 into x- + C into x -1 s equal 0 that's it it this is only the equation of the quadratic for which Alpha + 2 by Alpha minus 2 Beta + 2 by Beta minus simp minus x² minus Alpha + beta into x + Alpha betaal 0 Roots this and this x² minus sum of the roots X1 + X2 into X Plus product of the roots X1 into X2 equal Z so x111 Alpha by Beta plus beta by Alpha into X Plus X1 X2 Al by Beta betaal beta beta very good x² minus Alpha sare + beta s by Alpha Beta into x + 1 is Ro Al + beta square a b sare a s+ b 2 Bet square plus 2 a sorry minus 2 Alpha Beta plus 2 Alpha Beta sorry plus 2 alha Beta left 2 Al beta Al sare beta Square Alpha + beta whole Square minus B by a whole Square minus 2 * of Alpha Beta C by a so we got here b² by a sare - 2 C by a b² - 2 a by a a s a sare x² sare beta Square b squ - 2 a c by a square by Alpha Beta C by a C by a into x + 1al S here I'm going to write okay A C x² C a x² - b² - 2 a c into x + a c is equal to0 so this is the quadratic equation for which Alpha by Beta And beta by Alpha are the roots squ by Beta beta Square by Al next uh x² minus uh X = equ question number one alpha plus beta in number two alpha into beta in number three modulus of alpha minus beta in the number four Alpha square + beta square is what 5 Alpha Cube + beta Cub is what number six Alpha Cube uh minus beta Cub is what 7th one alpha power 4 + beta power 4 is what eth Alpha by Beta + beta by Alpha is what 9 Alpha Square by Beta + beta sare by Alpha is what 10th similarly 1 by Alpha + 1 by Beta is what the roots minus B by a 1 b 1 c 1 so sum of the roots minus B by Aus B byus so sum of the roots minus one c c Alpha minus beta square is nothing but Alpha + beta squ minus 4 Alpha Beta am I correct sing on both mod of Al minus beta is nothing but square root of alpha + beta square root of alpha + beta square is -1 square is + 1 - 4 * of Alpha Beta so this is square root of -3 we can write it as root3 into I3 I clear alare third question sare 1 square - 2 into 1 so which is 1 - 2 which isus one Al Square beta Square sir squares fifth one Al beta a a b Into A sare + B sare minus a so Aus A sare + B sare minus1 alpha beta 1 so this is min -1 into min-2 which is nothing but 2 so Alpha Cube plus beta cube is 2 CLE bet beta into Al bet bet bet Al Square beta Square minus1 sorry minus1 Al beta3 IUS Al beta 1 beta Alpha square square + beta square square Alpha square + beta Square square- 4 * of alpha square + uh- 2 * of alpha Square beta Square sir so Alpha square + beta square and Theus one sir -1 square into 2 * of 1 sir this is 1 - 2 which is min-1 Sir the answer Alp square + beta Square by Alpha Beta Alpha + beta by Alpha Beta Alpha plus very a x² + b x + c equal to0 then if if this is a quadratic then it has two roots B it has Ro is it possible [Music] for example sin squ x plus cos sare saref xal to 1 bet satisfied 2 Bet satisfied 100 bet satisfi P bet satisfi Z bet satisfi X Val satisf X belongs to all the real numbers it is correct so that's why it is not qu we call it as if a x² + b x + C is equal to 0 has more than more than two roots Ro then then a x² + b x + cal0 is called is called an identity that implies AAL 0 Bal 0 0 into x² + 0 into x + 0al 0 how many X values will satisfy this satisf x = 1 by2 s 0 into 1 0 0 into 1 0 0 + 0 0 0 equal 0 correct sir so x = 1 is a solution is a root 2 by two 0 into 2 square 0 0 into 2 0 plus 0 two is also a root 320 4 0 5 belongs to all the real numbers [Music] if and only condition okay if K Square sorry K Square minus 1 into x² + k s uh - 3 k + 2 into x + uh k s - 5 k + 4 is equal 0 has three Roots three roots then find K value then find the value of K is what and identity it must be identity it is an identity 3 = K value either 1 or 2- 5 k + 4 is K value final answer so what is the common K value common K is nothing but one so K is nothing but how one so one is only the correct answer values what- + 1 1 2 what is the common K value that is the final answer okay M 26th December results right Ro first of all A1 x² + B1 x + C1 equal 0 quadratic where A1 not equal to Z similarly a 2x² + b2x + C2 is equal Z where a not equal Z the two qus Quadra maybe root bet no common Roots no common Roots beta beta bet so we say that one common root two root two roots are common two Roots two common [Music] Roots single Comm Ro two Comm Roots okay right this is very simple number one if q1 and Q2 have a common root a common and a a means what one one common root a common root then minus B1 A2 A1 B2 A1 B2 minus B1 A2 B minus BC equal B c c c okay right 1 2 1 1 2 1 that's it A1 B2 minus B1 A2 B1 C2 minus C1 sorry B2 C1 C1 A2 minus A1 [Music] C2 Okay so 2 so A1 B2 minus B1 A2 B1 C2 - C1 B2 is equal to C1 A2 - A1 C2 whole Square that's it whenever these two quadratics are having a common root or one common root then this is the exact condition we should write if q1 and Q2 are having two roots in common two roots in then important very simple then a 1 by A2 equal B1 by B2 equal C okay right a x² + bx+ cal0 where a not equal to Z is the quadratic quadratic discriminant important discriminant root B 4 a 2 netive number Square discus B plus discriminant by 2us B disc if D is not a perfect square one is a perfect square 0er is also perfect square one is a perfect square four is a perfect square 9 16 25 36 49 squ s 1 4 square is not a perfect square then roots are then Roots root p - b + 0 by 2 Aus B- 0 by 2 A- 2us yes roots are and equal Roots do not exist are the roots are imaginary do not exist real exist not real do not exist not comp Roots roots are p+ i p i very very very very important qura expression a² BX number one first number one discriminant Z discriminant POS upward L downward qu POS roots are real and distinct roots are distinct Roots stinct distin Ro equal Roots no roots imaginary roots one number two number three three graphs first one in first one discriminant greater than zero discriminant tells about only the nature of the roots discri Ro so these are the total six types of graphs of the quadratic always greater Z quad okay so rever idty common roots so there is AAP exp for example for example the quadratic quadratic uh x² + 4x + K = 0 has a root root 2 - 3 I has a root 2 - 3 I then find then find K value K value comp p i 2 I3 2 second Ro should be should be the second Ro beta beta ro ro C by a so product of the roots root 2us 3 I root 2 + 3 I equal K so a square- b square a square- b² = k so a square and 4 minus b square and 9 i sare s i and square root of minus1 I sare so I sare into plus so K that's it very good for example if 2 + < tk3 is a root for x² - a x + uh Bal 0 then find a values a comma B is subscribe to the channel channel subscribe please where Q is not a perfect square sum of the roots Alpha plus beta is minus B by a a and product of the roots C so of Ro of roots ot3 2us root3 Roots b sir 2 + < tk3 into 2 - root3 sir a sare - b sare 4 - 3 which is B Val 4 1 very good super example if if x² + 5x + 10 b² - 4 is equal to0 has uh has this end and x² + PX + Q is equal to0 have a common root have a common root then find then find p² + qal how much 30 seconds time I'm giving you 30 seconds time so please can you solve the question question solve then find the value of p s 75 right 125 super Vishnu discrimin b² 25 minus 4 a 4 into a into C A 1 C 10 25- 40 which is -5 Ro imag IM qal a 1 by a 2al B1 by B2 C1 by C2 so 1 by 1al P by 5al Q by 10 sir so P = 5 qal 10 sir p² + Q ² is equal to 25 + 100 which is nothing but 125 statement FSE for example I have 100 rupes I have 100 rupees true FSE true false statement whether it is a true statement are false I have true or false true or false first very good super super super super yes the statement is true absolutely true same thing here also first Quadra second quadratic first qu second maybe second okay Maxim minim already function I told I told this in functions function a² b c maximum Min discus b s - 4 a c by 4 a so minimum point this is the minimum that point is minus B by 2 a comma minus Delta by 4 a Delta discriminant so X = only minimum maxus discriminant [Music] F Max right next location of the roots location of the root point there are six methods six methods number one location of the roots for this qu the roots are less than Ro but I'm unable to do location of the roots so hours complete that I can complete theab in time okay right are the roots for the quadratic equation Theta belongs to 0a by2 then this value is what x sin thetus 2 sin thet equal bet of the roots beta - B by a sin Theta product of the roots Alpha Beta plus C by a minus 2 sin Theta betus Sin thetus Sin Theta Al plus beta 2 * of alha plus beta I got this relation relation four of the any number of dice any number of Alpha power 12 + beta power 12 by 1 by Alpha power 12 power negative 1 by Beta ^ 12 into Alpha - beta whole power 2 bet beta power 12 + Alp power 12 by Alpha Beta power 12 into Alpha Beta whole power 12 uh 12 by alus Beta 24 next Alpha Beta Alpha minus beta so Alpha Beta 12 by alus Beta 24 Al beta by Al beta 24 Al minus beta sare sare beta Square minus 2 alha Beta yes Al beta by Alp + beta s - 4 Al beta I got that Alpha Beta is 2 * of alpha plus beta I can replace Alpha plus Alpha Beta with 2 times of Alpha Beta Alpha plus beta Al plus beta Square minus 4 * of four * of Alpha Beta 2 * of alpha plus beta answer 12 is equal to 2 * of alpha + beta by whole 12 cancel so this is what 2 by Alpha + beta - 8 12 beta Sin Sin thet so this is 2 by sin Theta - 8 and + 8^ 2 sin 2 Theta + 8^ 12 2^ 12 by sin Theta + 8 12 a option is the correct answer 2 12 by sin thet + 8 clear option option PQ are the real numbers 2 - root3 is a root of the quadratic equation for this then which of the following is correct Babu root2 2 let qu equation so qu equation x² minus sum of the roots 4 X Plus product of the roots 1 is equal to 0 so qu x² - 4x + 1 = but x² + PX + qal Q satisf correct answer p² minus 12 yes B option correct answer p s - 4 S - 4 into q and 1 - 12 which is 16 - 16 very good Zer is the correct answer B option b option is the if m is chosen in the quadratic equation such that the sum of its roots is greater greatest then the absolute difference of the cubes of its roots is what location of the roots Comm Roots just Roots question is chos in the quadratic equation such that the sum of its roots is greatest sum of the roots m² + 1 into x² - 3x + m² + 1 whole Square sum of the roots Al beta Roots sum of the roots Al plus beta minus B by a of the root when it will be greatest PC as right 4,000 step 4,000 steps are completed when it becomes greatest denominator minimum so it becomes it becomes greatest when when the denominator m² + 1 is minimum minimum minim so m² + 1 minimum is 1im so 0 + 1 1 so minimum in the denominator one so greatest of greatest value of value of alpha + beta is 3x 0 + 1 which is nothing but 3 so greatest bet m equal Z M okay then the absolute difference of the cubes of its roots alus beta differ alus beta into Al s + beta S Plus alha beta expansion so mod of Al minus beta product of the roots alha beta is C by a c by a and m² 1 but so bet bet beta is what square root of alpha + beta whole squ minus minus 4 * of Alpha Beta which is modulus of alpha minus beta so after that Alpha plus beta Alpha + beta whole squ minus Alpha Beta - 2 Alpha Beta sare root of alpha + beta is 3 Square 9us alpha beta 1 4 into Alpha + beta whole square and M the 9 minus Alpha Beta And 1 so this is is 8 into < tk5 is the correct answer 8un 5 the B option is the correct answer simple questions if bet are the roots of the quadratic equation then the least value of n for which this is equal to one alpha by Beta bet x² - 2x + 2 = x - 2x + 1 + 1 0 per x = 1 xus x = root1 i = 1 + r i = 1 I 1 I Al by Beta Al 1 + I by I this is the least Val least positive value then the least value of n n can be zero also but Z four is the correct answer C option very goodma super 2 i i 2 I very simp RO number of questions the number of integral values of M for which the equation has no real Roots 4 into 1 + 3 m b² - 4 into AC 1 + m² into 1 + 8 less than 9 M 6us 1us m² - 8 m - 8 m Cub less than 0o uh 8 m CU - 8 m 1 one gone - 2 m + 8 m² less than 0 so 4 m² plus 2 M 4 M - 1 M value am I correct so Roots M = minus b + Rus root of b² - 4 a c by 2 a 2 into 4 so m is = - 4 + orus 4 < tk2 by 2 into 4 4 and canc m equal -1 + orus < tk2 by 2 so so M number of integral values of M for which equation 1 2 -1 uh -1 + < tk2 by 2 -1 - < tk2 by so POS the number of integral values of M for which the equation so either correct answer common step common step wrong r ah okay okay okay Square - 4 m + 1 great than b s - 4 A- B plus orus root of b² 2 m into this is m 2 m - 1 s greater than zero so always positive so M must be greater than Z and M not equal to 1 by2 M greater than Z b² 1 + 3 m equation is this correct am I correct is this correct W asant option number of inte correct the equation VII 4us 4mi 9 m m 8 m 8 so once you also see it answer okay - 8 m - 8 m uh + 6 M - 2 m m s + 8 m sare less than zus Plus 4 m² - 4 m + so option b is the correct answer right it is 1 M or not option is the correct answer option right okay guys assment five questions 10 minutes so assignment assment okay so that's all for now guys by everyone and please like the video share the video with your friends and please subscribe to the Channel 100 likes that likes so definite integration indefinite are differential equations mat and DET okay right so so loation of the root question first of all okay guys so that's all for today now bye everyone thank you all thank you guys please like the video share the video with your friends and please subscribe to the channel thank you bye

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Id: UymR2V9mWi0

Length: 107min 37sec (6457 seconds)

Published: Mon Dec 18 2023