hi students I'm Iran and today we're going to discuss the past purpose related to the quadratics e level maass our first question is it says find the value of the constant C so C equals to question mark and for which the line Y = to 2x + C is a tangent to the curve which is y = to which is Y2 = to 4X so now we have two lines actually here this is our uh stra this is our line and this is our curve y = 4x we need to make sure that we have uh the value of the Y not the value of the square so for that reason we are going to take the square root on each side which will make it as Y = 2 into X square root or we can we can just keep it as it is which is 4X square root so y become equal 4X s root now the very first thing we need to we are given a straight line we are given a linear uh equation and we a quadratic equation and we have to find some value of C with the tangent to the curve is y tangent mean if it says tangent it means b² - 4 A C = 0 Z but before we work on that one we need the equation quadratic equation from which we can find this value and we can solve the answer so that reason we are going to equate them this is the first step as we uh discussed earlier so 2x + C equals to is going to be equals to a square Ro TK of 4X so at at the first stage we're going to take the square on each side so Square here and square over here this will be become 4x² expanding the square 4x² + c² + 4x c = 2 square root cancels by square which becomes 4X so let's shift it to the left hand side 4x² plus uh 4X c - 4x and + c² = 0 so 4x² plus uh let's take X common out getting the coent in only so 4 c - 4 into x + c² = 0 this is our final Quadra equation for this question so in this final quadratic equation which we have we can clearly see that our a value is 4 which is coefficient of x² B value is coefficient of x which is 4 C minus 4 and C value is actually c² now we know that if it's a tangent to the curve this is our curve uh this is our curve and the line is tangent to just intersecting a single point so where the b s the discriminant should be equals to zero for that so let's s the Bing b² is 4 c - 4 x² - 4 into a is 4 into C is C Square should be equals to zero everything so let's expand this Square so it's a 16 c^ 2 + 16 - 32 c - 16 c² = 0 16 + 16 c² - 16 c² they cancel each other and 16 uh - 32 C = 0 32 C shifted to the opposite side so 16 become equals to 32 C now dividing each side by 32 we get uh half equals to C so the value of C for this question becomes equals to half or 0.5 this was our first question so we had we have to find the value of C uh the equation for the line was 2x + C the quad equation was y^2 = 4X from where we got the we need the single y value not a square of Y value in order to get the square in order to get a quadratic equation so we got we took the square root on each side y become equal to 4X now this is our quadratic equation and this is our linear equation the first step is we always equate them first after equating we simplify them into a quadratic equation once it's a quad equation we look at the information given they said the line is tangent to the curve tangent mean b² - 4 which is discriminant it is it should be equals to zero so finding the b square - 4 is value and we actually simplify it with the value of C question number two okay in the second question it says find the real roots of the equation real Roots they did not mention the real roots are uh is a positive value or or what but they just said find the real Roots okay find the real roots of the equation the equation is given 18 over x^ 4 + 1 / x² = 4 and we have to find the real Roots we real Roots mean the value of x actually so let's simplify it first uh it's uh 18 / X4 + 1 / x² so we have to multiply the this equation by x² here and X squ here so we can get the same denominator so equals to 4 then we have the common denominator become equals to X4 and this is 18 as + x² = 4 let's multiply x 4 to the opposite side so 18 + x² become = to 4 * x^ 4 now let's shift everyone to the same side so we're going to shift the uh these values to the opposite side here so it is going to Beal to 4X 4 - x² - 18 = 0 we're dealing with Quadra equation not uh the by Quadra equation so for that reason we're going to let let a = to x² and then we when we take the of each side it will become a² = x^ 4 so now we have these two values and we're going to substitute into the equation so it will become uh 4 into X4 mean a s - x² mean only a then - 18 = 0 this become our simple quation can do it by factorization or we can apply a uh quadratic formula so I'm apply I'm going to apply a quadratic formula here so a = to - b + - b^ 2 - 4 a c / by 2 a this our unknown value better take another letter I think that would be better okay soal to minus B value our B value is minus a let me write the values a = 4 B = -1 and C = to uh -8 18 so minus B minus b means minus into -1 + minus b s again so Min - one perfect - 4 * a a is 4 and C is -8 everything divided by 2 into a and a is 4 so let's simplify it further 1 + minus okay so we have -1 per sare 1 - - plus 18 * 16 18 * 16 that's 288 this gave us uh this gave us the value of 288 so 288 / by 8 so let's see if we can have the square root of 289 so square root of 289 becomes 17 s so = to 1 +us uh 289 everything divided by 8 so we are either having a equal to 1 + 17 / 8 or we have the value uh 1 - 17 over 8 if if it is 1 + 17 / 8 it will become 18 over actually the value so missing the space right now color so so the first say is actually this first value become equals to 18 / 8 18 / 8 which is actually equals to 2 9 * 18 to 4 * 8 okay and the second a value become equals to 1 -7 which is -6 / 8 and it will become -2 eventually so these are our two values for a but we know that the a value was actually equals to x² so when we say that a is x² actually so our first this value will become x² = to 9 by 4 and another X = to -2 uh Min -2 here because we switch a with X so X = to min-2 so when we say x = 2 uh sorry x s x s minus 2 so how can we get rid of the X squ we are we going to take the square root on both sides for this one and the for other one as well so taking square root on each side here and here as well so this side will become X cancel by x x become equals to square root of 3 by uh 9 by 4 will become plus minus for 9 it is three and for four it is two so this is the this is one of the value and for other value here x will become equals to now Min - 2 sare root minus 2 squ root the questioner says the questioner says find the real roots of the equation this is not a real root X is minus 2 perfect square root minus square is not a real Ro it's an imaginary number so we're going to skip this one we're not going to select this so our answer becomes x = + - 3 / 2 that's how we can find the real Roes when we were given an equation here let's go to the next question it says determine the set of values we have to find set of values set of values equals to question mark of the constant K uh we have to find the value of for the constant K okay this is our third question for the line line is y = 4x + K does not intersect the curve and curve is y = x² this line is not intersecting this one not intersection means what not intersection means uh b² - 4 a c is going to be less than zero that's what meant by not intersection not intersecting there's no intersection between okay so first of all same method first of all we have to equate the two equation so 4x + K = x² shift everything to the right hand side so x² - 4x - K = 0 afterwards we uh find our a b and C's value so here the a value is 1 and B = to -4 coefficient of x and C is the constant term which is minus K in this case as we just knew that uh it's not intersecting mean the b s - 4 a c is going to be a negative number so b^ 2 - 4 a c going to be a negative number B is -4 per s - 4 into a is 1 into C is minus K and everything is less than zero so it's 4 4 * 16+ 4 K less than zero we can shift the 16 to opposite side to find the value of K which is going to be equals to uh less than - 16 dividing each side by four so K is less than n minus 4 then answer in inequality actually this is called set of values answer in inequality is called set of values so we were given a uh equation for a linear equation quadratic equation we equate them first turn it into a quadratic equation then we look at our discriminant no intersection was between them so we predicted that b s - 4 C must be less than zero or a negative number we uh substitute the value into this expression and that's how we could find the set of values for K question number [Music] four question number four says find the set of values of K again so have to find the set of values again set of values of k equals to question mark for which line Y equals to right here line Y equals to ax - 4 intersects the curve and the curve line equation is y = to x^2 - 2x and the intersection is at two distinct points two distinct points intersection is that two distinct points whenever it says two distinct points it mean your discriminant is positive b sare - 4 a c is greater than Z or positive so same thing first step equate the two equations so KX - 4 = x^2 - 2x shift everything to the right hand side so X2 - 2x uh - KX + 4 and should be equals to everything zero let's uh take the X common out so we can have the coefficient of x which is 2 + K into x + 4 = 0 now we have the b s - 4 a Val greater than 0 so let's write this b ^ 2 - 4 a c greater than zero so let's find the valine from here of B value is min - into 2 + k perx s - 4 into a is 1 and C is four and everything should be a positive so when we take the square of the negative it becomes positive and when we expand 2 plus K it will become 4 + k s a s + b s + 2 a or 4 K - 4 4 * 16 greater than zero we are going to have k 2 + 4 K - 16 + 4 it will become -12 greater than 0 this is our quadratic equation which we had out of the two now we're going to to factorize this so 12 means either it's 1 * 12 2 * 6 3 * 4 or 6 time so it will be repeated again so we need four and this will give us four here actually so two and six factors can give us 4 so k 2 + uh 6 K - uh - 4 - 2 K - 12 greater than zero okay then we have two pairs so from the first pair we have K Square uh K common out so it will become K into k + 6 minus from the second 2 is common so - 2 into k + 2 6 * 12 everything greater than zero and we have k + 6 common into K - 2 greater than 0 what to do next okay once we have this equation we're going to draw a graph our 1K value is if we assume that k + 6 = 0 so then K becomes minus 6 this is our first point and if K - 2 = Z then K will become equals to 2 for us so we have here somewhere here as - 6 just a rough sketch and we have a positive two value okay it's greater than zero and it's a it's a positive it's a positive coent so it's a smiley face by the way so let's make a smiley face here okay when come to the sharing you can see so greater than zero mean we are going to share the graph above xais so now we can write the set of values very easily our K value exists Which is less than - 6 so our K value is K value is less than - 6 from the left hand side and for the right hand side the K value is greater than two so we are moving this way and the values are actually getting bigger and bigger than two so K here is greater than two so we were given we were to ask the valoc set of K we were given a line a qu equation intersec at two distinct points that's how actually it look like so we uh simplified the equation having b s - 4 C greater than zero we simplified it and then we found our K values we draw and we label them we color them and then we find the set of values that's question number four question number five it says determine the set of values of K so set of values set of values of k equals to question mark and for which the line our line is uh 2 Y = X + K we need only y so we shift to the opposite side so it will be x + k / 2 this is our line equation and it does not intersect the curve does not okay there's no intersection with the Curve and the curve equation is y = to x² - 4x + 7 and the two are not intersecting does not does not intercept does not intersect this is important when the two graphs does not intersect together it means the value is a negative Val of the discri so B sare minus 4 a c is going to be less than zero in this case Okay so same step first of all we equate the two equations so x + k / 2 is going to be equal to x² - 4x + 6 shift 2 to the opposite side so x + K = 2x² - 8 x + 14 now shift x + K to the uh same side so we can get the final quad equation 2x² - 8x - x + 14 and - K = to uh is uh going to be yeah equals to Z let's simplify it further 2x² - 8 x - 9x + 14 - k = z our a value is actually 2 B value become - 9 and C value equal to 14 - K discriminant no intersection between them so it is going to be a negative or a less than zero so b^ 2 - 4 a c should be less than zero B is - 99 so b 2 - 4 A and C 14 - K should be less than Z let's simplify them 9 I'm 81 - 8 into 14 8 into 14 it becomes 112 So 8 into 14 112 and minus minus AS Plus 8 into K 8 K less than zer okay let's simplify the like terms together so 112 minus 81 this will give us 31 here with the negative sign - 31 + 8 K less than 0 that's further simplifi shift 31 to the opposite side it will become 8 K less than 31 or k is less than 31 by 8 so we were to find the set of values of K uh for a given line and a quadratic equation which does not intersect each other does not insect means the value is negative for the discriminant We equated Them first then we simplified them and turned them into a quadratic equation find found the ABCs of them then use a uh given discriminant by the definition simplified it and eventually we got the Val question number six a curve has equation a curve has equation Y = to Y = to K x² + 1 and the line has equation Y = to KX we our quad equation this is our line equation where K is a nonzero constant okay k for the K condition of K they said k equals to non zero constant other than zero it could be any value okay the first question first question they said find the set of values of K for which the curve and the line have no common points so set of values set of values of a we have to find them and information is does not intersect does not intercept does not intercept mean the value of the discriminant must be a negative number B first should be less than zero okay same thing first equate them so K x² + 1 = KX let's shift it to the left again K x² - KX + 1 = 0 okay so here we have a = K B = minus k c = 1 so B is - k 2 b² - 4 A and C is 1 it should be less than Zer less than 5 k s - 4 K should be less than zero let's shift minus 4K to the opposite side it will become k sare uh a s minus 4 K if we shift it to the opposite side we have to take the square root which is not allowed we cannot even do the either or here uh so just take K common so we have into K into K - 4 less than 0 [Music] so we are going to assume if k equals to 0 and K = to 4 then we draw a graph so it's a graphical explanation graphical solution we have k equals to Z and we have K as 4 okay what type of uh coefficient of K is coefficient of K is positive so it's going to be a smiley face it's going to be a smiley face coent of K is positive so smiley face okay it's less than zero less than zero mean you have to color the area color the graph under the x-axis so we just colored it now what is this range let's try the range K is greater than z k is greater than zero okay and K is less than four K is less than four if we combine them together 0 is less than K and K is less than 4 this will become our final answer for question number six part one in the second part it says State the Valu of K for which the line is a tangent to the curve and for this case find the coordinates of the point where the line touches the curve okay so we have a different condition here let me write the condition he say the set of values of K we have to find set of values of k equals to question mark line is tangent tangent line mean touching only at one point and one point mean b^ 2us 4 a going to be equals to zero Bally and for the case find the coordinates of the point where the line touches the curve we have to find the coordinates where the line touches the curve this we need to know these coordinates of this point as well so first of all Let's uh equate the two equations as usual which we already did in the in the last question which is K x² K x² uh minus KX and + one equals to zero okay our B S 4 is equals to Z actually so B is minus k b 2 - 4 A and C = 0 also k k sare - 4K = 0 or k ² = 4K k s = 4K now we have a different condition we have equal sign so we can definitely use the either all condition for us here okay so instead we should uh just take K common here it will be more easy then okay K into K - 4 = 0 now either or k = to0 or k = 4 okay two conditions now they said the key value should be a nonzero constant so we're going to reject this one this one will be rejected because it doesn't fulfill the condition but k = 4 is going to be an answer okay for this case find the coordinate of the point where the line touches the curve we got the value of K we got the value of k equal to 4 k = 4 now we have to find the coordinates of the points which touches the which touches line and and the curve are actually intersecting together so value of K if4 then our equation is going to be now K is 4 so 4x² - KX which is 4x + 1 = Zer now if we find if you can simplify this we can find we can get the x coordinate and once we get the x coordinate we can easily substitute the value of x into either of the equation and we can get that point as well so let's simplify it for uh so it's going to be equals to okay uh that's basically a s so 4 x² - 2 a so it's going to be - 2x - 2x + 1 = 0 or uh 2x common here so we have 2 into x -1 minus nothing common one should be considered there's the next sign sign of inside change so 2x - one = 0 same numbers so we are doing it correct 2x - 1 into 2x - 1 = Zer so our k for either R we can find the value of x become equals to - 1i 1 by 2 this become the value of x which is actually repeated if x coordinate is half then what is the y coordinate for y coordinate you can use any of the equation uh for example I use this equation so y = to K is 4 x is half so this will become two so our coordinate where the two are intersecting each other x coordinate is half and the y coordinate is two that's how you find the coordinates of the point where the two are intersecting [Music] together question number seven question number seven the equation x² + p x + Q = 0 there is an equation given which is x² + PX + Q = 0 where p and Q are constants okay has roots minus 3 and five Roots mean value of x is min -3 and five there are two values of X it says find the values of p and Q that's the first question what is the value of p and what is the value of Q what to do now well we have values of x given so let's subtitute into the equation so x² + PX + Q = 0 if x is min -3 for example we are using x = minus 3 first so this will become Min - 3x² + P into -3 + Q = 0 so this will become 9 - 3 p + Q = 0 okay we can we can Sol it actually by a comparison or a simultaneous method kind of so I want to send everyone other side except for Q so Q will become equals to 3p - 9 this is one value here while x = minus 3 what if x = 5 is considered now then our same equation will become x² + PX + Q = 0 will become as X is 5 so 5 perect + P into 5 + Q = 0 or 5 5 * 25 + 5 p + Q equals to Zer I'm again going to find the value of Q here so everything the opp side so Q will become equals to - 5 P - 25 both are equals to Q so it means we can equate them so 3 p - 9 can be equal to - 5 P - 25 now like terms together so 3 p + 5 P = -25 + 9 or 8 p equals to - 25 and 9 it will become -6 or B become equals to - 16 by 8 or -2 so this will become the value of for p -2 if p is min-2 then what is the value of Q we can use any of the equation for example I'm using this equation so value of the cube become equals to 3 p - 9 or 3 into P we found -2 - 9 = - 6 - 9 or -5 so the value for the Q become equals to -5 that was the first part of the question now the second second part of the question says using these values of p and Q find the values of the constant R for which the equation has equal Roots equal means equal it has uh basically B sare going to be equal to zero so let me raise this first and finding the Val the value of B we found equals to -2 and Q we found equals to -5 so let's move one for the second part for the second part is said for the second part they said uh using the Val of p and Q uh find the value of the constant R R equals to question mark for which the equation there's an equation given here x² + PX + Q + r equal to0 has equal roots and equal Roots mean the b s - 4 a c uh equals to Z so equal Roots it has equal Roots the equation has equal roots and equal Roots mean what equal Roots mean b² minus 4 is going to have a single value so it's going to be equals to zero here single answer B for C equals to Z x² question of X and C we can simply start following the b² - 4 a c so b s - 4 A C = 0 B is p perf s - 4 into uh a is 1 and C is Q + r should be equals to Z Now p s is what p sare mean - 2 per s - 2 per s - 4 1 4 into Q is -5 so -5 + r = to 0 this will become 4 - - + 15 4 * 60 - + - 4 R = 0 or 64 - 4 R = 0 now 4 R = 64 or R = 64 / 4 which is 4 1 * 4 and 4 6 9 24 so R's value is actually going to Beal to 16 next question question number eight if you guys have any difficulty in the question pause the question go through it again completely and then switch to the next question so question number eight next find the value of M so find the value of M so we are finding value of M values of m equals to question mark for which the line equation of the line given y = mx + 4 intersects the curve it intersects there is an intersection y = 3x² - 4x + 7 at two distinct points so they said two both are intersecting at two distinct points two distinct points when it says two distinct points is actually greater than zero so this is b s - 4 a c is greater than Z two distinct points b c greater than Z okay same thing equate them first so mx + 4 should be = to 3x² - 4x + 7 sh everything to the right hand side so 3x² - 4x + 7 - MX - 4 = 0 or 3x² - 4x - uh MX will become uh basically we will take the X common so it will become 4 + m into x + 7 - 4 that's 3 = 0 okay so here we are values b² - 4 a c uh two distin points so it should be greater than zero B is minus into B is minus into 4 + m that's b 2 - 4 a is 3 and C is three as well a three C is also three yeah so it's going to be greater than zero so let's simplify it now minus will become positive 4 become 16 + m² + 8 m minus uh 3 * 9 94 * 36 greater than zero so m² + 8 m - 36 + 16 that's minus uh 0 and 2 greater than Z this is our quadratic equation right now for this quad equation we're going to solve it now so we solve it for 20 20 can be 1 * 20 or 2 * 10 or 3 4 * 5 5 * 4 are 10 * 2 so it's already repeating so which one is going to give us 8 here this one 2 and 10 okay so m sare + 8 m - 20 is greater than Z or m² 2 and 10 so it is + 10 m - 2 m - 20 greater than zero or m² + 10 m - 2 m + 8 m - 20 greater than Z so sorry okay so we have two pairs here so from this one m is common so m + 10 - 2 into m + 10 Again greater than zero m + 10 into M - 2 greater than 0 now we're going to get the help of the graph here in this case so we are either having M = - 10 or m = 2 graph we have Min - 10 on the left hand side so minus 10 positive on right hand side so plus two gra is a positive m s is positive so it's going to be a smiley face it's going to be a smiley face okay it's greater than zero so we have to label the graph over the x-axis not under it so this will give us the value of M value of M will be equals to uh great less than minus 10 m is less than minus 10 and here m is greater than 2 m is greater than two so that's how we can find the value of M here question number nine we should solve at least 10 questions today it says the line there's a line given y = to X over k + K where K is constant is a tangent to the curve tangent tangent to the curve tangent mean only a single point and single point mean b² - 4 a c is equal to 0 okay the curve line is uh y = to it's basically 4 y = x² so shifting for OPP sides y = x² over 4 these are our two equations let's equate them first now so X over k + K = x² / 4 let's simplify this time K time K so this will become K here in the denominator position and here we have x + a² = x² / 4 Let's cross multiply them so it become 4x + 4 k 2 = KX Square okay uh everything to the left hand side so K x² - 4x + 4 a² equal to0 all right so now we're going to uh this is our final equation kx2 - 4x and gu also shifting so it's minus 4 KX 4 K squ okay we got the a BCS here a = to k b = -4 C = minus uh 4 k s that's C okay so when we do that we have tangent so b s = to Z so B is -4 x² - 4 into a is K and C is - 4 k s uh- 4 a c that's or C - 4 a c = z this - 4 becomes 16 - - plus- - plus 4 4 * 16 K * k 2 that's K Cub = 0 shift 16 toid so it becomes 16 K CU = to - 16 I divide both size by 16 K CU K CU = to minus one by the way minus one so when take the cube root on each side so K will become equals to uh minus1 because cube root of anything cube root of one or minus one is actually same as you so we just found the value of K now in the Second Step they said the coordinates of p as well coordinates of the P it's a t so this is a curve this is T this one is point V and they said find the coordinates as well so how can you find the coordinates as usual we can find the coordinates by uh substituting value of K here first this one so equation first K x² - 4x - 4 k 2 = 0 so K Val minus1 so -1 into x² - 4x - 4 into k 2 -1 of x² = 0 so this will become - x² and - 4x and this will become positive so - 4 = 0 so - k 2 - 4x - 4 or we can switch their sign as uh x² + 4x + 4 = to 0 so this is our a s + b s + 2 a so 2 4X so we can we can write this as a in the in the perfect square form as a plus b perfect squ = 0 or x = -2 now if x is minus 2 for the point K what are the y coordinate Y coordinates are given in the example for example x² 4 so y become equals to x meanus 2 perfect squ and divide 4 so y become equals to- 2 per becomes 4/ 4 = to 1 that's how we found the value of K when the two lines are tangent so the coordinates of V will become X is- 2 and Y is 1 that's how we can find and this was question number SC okay question number nine let's do the last question question number 10 question number 10 last question of our today line Y = to x/ K plus sorry same question repeating okay the straight line Y equals to mx + 14 equation for a straight line is tangent to the curve it's tangent to the curve so tangent mean uh is only intersecting at one single point and in that case the b s - 4 a c = to0 okay and uh to the curve y so curve y equal to 12 /x + 2 at the point P find the value of the constant M and the coordinates B so have find the M value and the constant as well so for very first thing we need to equate the two together so our m x + 14 = to 12x + 2 now let's simplify this first one so we have to multiply each one of them by X so multiply by x * x * X and * X so this will become M x² + 14x = to uh X cancel x 12 + 2x everything to the same side so M x² + 14 x - 2x - 12 = 0 m x² so this will become 12x 14 - 2 12 x - 12 = 0 we got our simplified quadratic equation it's a tangent so b s - = 0 so B is 12 b² - 4 into a into -2 = 0 or it's 144 - - plus 12 4 48 M = 0 now we shift 140 the opposite side so 48 m = - 144 or M = - 144 divided by 48 we can further simplify it 144 by 48 let's see if it's divisible 144 ided 48 so answer is three only so it's equals to -3 it means M = to- 3 all right so let's serve the value of M into the equation uh this one so Val of M we get the MX2 + 12x - 12 = 0 m stands for -3 x² + 12 x - 12 = zero okay again we had a quadratic equation we need to simplify it further okay so if I take the minus 3 common here it will be x² uh - 4x uh - 4x and minus will become + 3 4 * 12 = 0 so on each side we divide minus 3 so we get the root of this so we are with x - 4x + 4 = to 0 which is equals to x - b b value is 2 a value is X x - 2 per simp equals to Z Now how can we get the gra of square we take the square root on each side which which will give us x - 2 = 0 or X = to 2 our answer is x = 2 yes x coordinate of the point what is the y coordinate y coordinate = to mx + 14 y = mx + 14 or Y = to uh value for m is - 3 so -3 x value is 2 + 14 so y = - 6 + 14 or Y = uh 8 so the coordinates of the point P become equals to 2 comma [Music] 8 this was the last question in we will be continuing into the next session and we'll solve more question from quadratics till then bye