EX 2.2 Theory & Examples 10th Math New Book NBF Sir Waqar Hussain
Waqar Well Wisher
[{'start': '0.480', 'end': '6.480', 'text': 'Greetings.  I greet all the viewers and'}, {'start': '3.439', 'end': '9.200', 'text': 'listeners in Waqar Velvashar'}, {'start': '6.480', 'end': '12.559', 'text': 'and start my class with Hadith Kareem.  The Holy'}, {'start': '9.200', 'end': '16.400', 'text': 'Prophet (peace be upon him)'}, {'start': '12.559', 'end': '20.000', 'text': "said that whoever takes people's property"}, {'start': '16.400', 'end': '24.240', 'text': 'with the intention of paying back a loan,'}, {'start': '20.000', 'end': '27.519', 'text': 'Allah will pay on his behalf.'}, {'start': '24.240', 'end': '30.960', 'text': 'And if anyone takes without giving, then Allah'}, {'start': '27.519', 'end': '34.320', 'text': "Ta'ala will also destroy him.  Dear children,"}, {'start': '30.960', 'end': '36.480', 'text': "in today's class we will discuss all the theory related to exercise 2.2"}, {'start': '36.480', 'end': '40.640', 'text': 'along with its examples.  So before starting the video,'}, {'start': '38.559', 'end': '42.719', 'text': 'please tell us in the comment section'}, {'start': '40.640', 'end': '44.960', 'text': 'from where are you watching our video'}, {'start': '42.719', 'end': '47.320', 'text': "and which college do you study in.  So let's"}, {'start': '44.960', 'end': '50.719', 'text': 'move towards our questions.'}, {'start': '47.320', 'end': '54.160', 'text': 'Equations Reduced to Quadratic'}, {'start': '50.719', 'end': '55.760', 'text': 'Form.  Even equations are not in quadratic'}, {'start': '54.160', 'end': '58.399', 'text': 'form.  Like in quadratic form we have the'}, {'start': '55.760', 'end': '61.520', 'text': 'standard equation ax² + bx +'}, {'start': '58.399', 'end': '63.760', 'text': 'c but using some substitution we can'}, {'start': '61.520', 'end': '66.479', 'text': 'reduce them into the quadratic form. By'}, {'start': '63.760', 'end': '67.840', 'text': 'using some substitution'}, {'start': '66.479', 'end': '69.760', 'text': 'we get a simpler equation which is'}, {'start': '67.840', 'end': '71.840', 'text': 'not in quadratic form.  We can'}, {'start': '69.760', 'end': '74.000', 'text': 'easily discuss it in quadratic form.'}, {'start': '71.840', 'end': '76.799', 'text': 'One by one we'}, {'start': '74.000', 'end': '79.840', 'text': 'also discuss its types of equations.  Type one'}, {'start': '76.799', 'end': '86.080', 'text': 'type number one.  In type one, the first'}, {'start': '79.840', 'end': '87.920', 'text': 'type that you have is ax 4 + bx² + c = 0. Now this'}, {'start': '86.080', 'end': '89.840', 'text': 'equation is'}, {'start': '87.920', 'end': '92.640', 'text': 'not in quadratic form.  But if we'}, {'start': '89.840', 'end': '94.159', 'text': 'try to solve it through some substitution'}, {'start': '92.640', 'end': '96.159', 'text': 'then we can easily convert it into quadratic form.'}, {'start': '94.159', 'end': '98.560', 'text': 'So in this we will do that'}, {'start': '96.159', 'end': '104.000', 'text': 'let'}, {'start': '98.560', 'end': '107.000', 'text': 'x² = y then square both the sides.'}, {'start': '104.000', 'end': '109.720', 'text': 'Squaring on both sides will give this:'}, {'start': '107.000', 'end': '116.000', 'text': 'x² ='}, {'start': '109.720', 'end': '118.880', 'text': 'y² 2 * 24 This will give this: x power 4 = square of y'}, {'start': '116.000', 'end': '120.799', 'text': 'Now, when we put these values'}, {'start': '118.880', 'end': '123.119', 'text': 'here, if we'}, {'start': '120.799', 'end': '126.719', 'text': 'try to convert it, we'}, {'start': '123.119', 'end': '130.879', 'text': 'can easily convert it.'}, {'start': '126.719', 'end': '134.280', 'text': 'This will come to a instead of x power 4 we can write'}, {'start': '130.879', 'end': '138.640', 'text': 'y squared this will come to ay'}, {'start': '134.280', 'end': '145.360', 'text': 'squared plus b instead of x² we can write'}, {'start': '138.640', 'end': '149.840', 'text': 'y plus c = 0 So there you have it, we have'}, {'start': '145.360', 'end': '149.840', 'text': 'reduced in terms of the quadratic'}, {'start': '150.920', 'end': '153.920', 'text': 'equation'}, {'start': '154.680', 'end': '158.239', 'text': 'reduce quadratic'}, {'start': '158.280', 'end': '163.840', 'text': 'equation this is where you have type one of it.'}, {'start': '162.000', 'end': '166.480', 'text': 'Now let us solve a similar example.  We'}, {'start': '170.360', 'end': '177.239', 'text': 'will solve it.  The question you have'}, {'start': '173.200', 'end': '178.920', 'text': 'is 9x^4'}, {'start': '177.239', 'end': '184.640', 'text': '-'}, {'start': '178.920', 'end': '186.560', 'text': '8x² - 1 = 0 Now this equation you have is'}, {'start': '184.640', 'end': '188.159', 'text': 'not in quadratic form.  We'}, {'start': '190.159', 'end': '199.959', 'text': 'can easily convert this into quadratic form using substitution.  Let x² = y take the square of both sides.'}, {'start': '201.200', 'end': '203.840', 'text': 'Squaring'}, {'start': '204.200', 'end': '213.519', 'text': 'on both sides will give us whole square of x²'}, {'start': '208.840', 'end': '218.159', 'text': '= square of y 2 * 2r4 This will give us'}, {'start': '213.519', 'end': '220.400', 'text': 'x4 = square of y. Put these values'}, {'start': '218.159', 'end': '222.319', 'text': 'in the equation.  So if'}, {'start': '220.400', 'end': '225.319', 'text': 'we put these equations in the main equation then'}, {'start': '222.319', 'end': '226.840', 'text': 'it will come to you like this that'}, {'start': '226.840', 'end': '236.360', 'text': 'instead of 9 x power 4, you will get'}, {'start': '232.280', 'end': '244.239', 'text': 'y² minus this will again come to you'}, {'start': '236.360', 'end': '248.519', 'text': '8 y - 1 = 0 so it will come to you that the'}, {'start': '244.239', 'end': '248.519', 'text': 'reduced quadratic equation is'}, {'start': '252.159', 'end': '256.160', 'text': 'not helping us solve this question further.'}, {'start': '254.239', 'end': '258.560', 'text': 'For now we will just understand its quadratic'}, {'start': '256.160', 'end': '260.320', 'text': 'equation steps by steps.  Like'}, {'start': '258.560', 'end': '262.560', 'text': 'in the exercise, if further question is'}, {'start': '260.320', 'end': '264.160', 'text': 'asked to us to solve, then we will solve it there'}, {'start': '262.560', 'end': '266.080', 'text': 'also.  But for now we will just'}, {'start': '266.080', 'end': '271.759', 'text': "try to reduce the equation in such a way that it converts into quadratic form.  Let's"}, {'start': '268.320', 'end': '274.639', 'text': 'move on to type two.'}, {'start': '271.759', 'end': '276.800', 'text': 'Type two.  In this, the'}, {'start': '274.639', 'end': '279.000', 'text': 'general format of the type of equation that you will have will be'}, {'start': '276.800', 'end': '285.199', 'text': 'that a'}, {'start': '279.000', 'end': '289.840', 'text': 'px + b / p of x = 0, so in this, whatever we will let, we will let'}, {'start': '285.199', 'end': '292.080', 'text': 'p of x = y, that is,'}, {'start': '289.840', 'end': '294.639', 'text': 'we will let the p of x equal to y.'}, {'start': '292.080', 'end': '297.280', 'text': 'If we put this y in place of this p of x,'}, {'start': '294.639', 'end': '299.400', 'text': 'you will get'}, {'start': '299.400', 'end': '308.960', 'text': 'ay plus b / y = c Now the next thing we will do is to'}, {'start': '306.400', 'end': '312.440', 'text': 'eliminate this y from the denominator, we will'}, {'start': '308.960', 'end': '314.960', 'text': 'multiply y by the most.'}, {'start': '312.440', 'end': '320.160', 'text': 'Multiply'}, {'start': '314.960', 'end': '322.919', 'text': 'y on both sides, so it will come to you y * ay'}, {'start': '320.160', 'end': '330.639', 'text': 'plus y *'}, {'start': '322.919', 'end': '334.160', 'text': 'b / y = y * c So this will come to you the'}, {'start': '330.639', 'end': '339.039', 'text': 'square of ay will cancel out with it'}, {'start': '334.160', 'end': '343.960', 'text': 'plus b is equal to c So it will come to you the'}, {'start': '339.039', 'end': '348.800', 'text': 'square of ay plus c going here'}, {'start': '343.960', 'end': '352.080', 'text': 'minus c plus b is equal to zero. This is the'}, {'start': '348.800', 'end': '354.639', 'text': 'equation that you have got in'}, {'start': '352.080', 'end': '358.840', 'text': 'terms of quadratic form.  Let us'}, {'start': '354.639', 'end': '365.720', 'text': 'discuss a similar example of this type'}, {'start': '358.840', 'end': '367.880', 'text': 'that if you get 2 * x² +'}, {'start': '365.720', 'end': '375.360', 'text': '1 -'}, {'start': '367.880', 'end': '378.840', 'text': '2 / x² + 1 = 3, then first of all if we let'}, {'start': '380.039', 'end': '389.440', 'text': 'x² + 1 is equal to y, then in this equation,'}, {'start': '387.039', 'end': '393.039', 'text': 'wherever x² + 1 comes, we will put y there.'}, {'start': '389.440', 'end': '398.319', 'text': 'So this will come to you'}, {'start': '393.039', 'end': '403.120', 'text': 'too.  Instead you will get y - 2'}, {'start': '398.319', 'end': '406.120', 'text': 'divided by y = 3 then multiplied by everything.'}, {'start': '403.120', 'end': '409.520', 'text': 'y'}, {'start': '406.120', 'end': '409.520', 'text': 'Multiply y'}, {'start': '410.520', 'end': '424.479', 'text': 'on both sides so it will come out'}, {'start': '414.680', 'end': '426.960', 'text': '2y * y - 2 / y * y = 3 * y This y cancels out with this y, so'}, {'start': '424.479', 'end': '428.280', 'text': 'multiply them with each other.'}, {'start': '428.280', 'end': '435.639', 'text': '2y² minus two will come here. 3y will shift it here.'}, {'start': '433.039', 'end': '442.240', 'text': 'Here it comes out to be'}, {'start': '435.639', 'end': '445.080', 'text': '2y² - 3y - 2 = 0. So there you have it, we have'}, {'start': '442.240', 'end': '448.080', 'text': 'converted it using substitution.'}, {'start': '448.440', 'end': '454.400', 'text': 'Reducible Quadratic Form.'}, {'start': '452.080', 'end': '455.759', 'text': 'This can be further solved in terms of another variable y.'}, {'start': '454.400', 'end': '457.759', 'text': "But for now let's just"}, {'start': '455.759', 'end': '459.680', 'text': 'look at the concept that if you have an'}, {'start': '457.759', 'end': '461.360', 'text': 'equation that is not in quadratic form, then'}, {'start': '461.360', 'end': '467.919', 'text': 'how can we convert it into quadratic form by substitution'}, {'start': '464.039', 'end': '470.080', 'text': "?  Let's move towards its third type."}, {'start': '467.919', 'end': '473.360', 'text': 'Type Three'}, {'start': '470.080', 'end': '475.680', 'text': 'Reciprocal Equation.  An equation that is'}, {'start': '473.360', 'end': '479.039', 'text': 'not affected by the replacement of a'}, {'start': '475.680', 'end': '481.599', 'text': 'variable by it reciprocal then it is'}, {'start': '479.039', 'end': '484.080', 'text': 'called reciprocal equation.  That is, if we'}, {'start': '481.599', 'end': '486.080', 'text': 'replace x with 1 / x and there is no'}, {'start': '484.080', 'end': '488.639', 'text': 'change in the equation, then you have what is'}, {'start': '486.080', 'end': '491.120', 'text': 'called a reciprocal equation.  Now this'}, {'start': '488.639', 'end': '491.960', 'text': "is its type.  Let's discuss this.  You"}, {'start': '491.120', 'end': '495.599', 'text': 'have'}, {'start': '491.960', 'end': '497.000', 'text': 'ax4 + bx'}, {'start': '495.599', 'end': '503.199', 'text': '+'}, {'start': '497.000', 'end': '507.759', 'text': 'cx² + bx + a = 0. The first thing you have to do is'}, {'start': '503.199', 'end': '509.400', 'text': 'divide x² on both sides.'}, {'start': '507.759', 'end': '511.479', 'text': 'This will come to'}, {'start': '509.400', 'end': '514.200', 'text': 'ax4 by'}, {'start': '511.479', 'end': '516.279', 'text': 'x² +'}, {'start': '514.200', 'end': '518.919', 'text': 'bx3 by'}, {'start': '516.279', 'end': '520.680', 'text': 'x² +'}, {'start': '518.919', 'end': '529.600', 'text': 'cx² /'}, {'start': '520.680', 'end': '531.600', 'text': 'x² + bx / x² + a / x² = 0 Now here,'}, {'start': '529.600', 'end': '533.320', 'text': 'if we cancel this x² with it, then'}, {'start': '531.600', 'end': '538.480', 'text': 'two will be left here.'}, {'start': '533.320', 'end': '541.760', 'text': 'x² is x then one will remain.  This will come to bx, after'}, {'start': '538.480', 'end': '543.480', 'text': 'cancelling with c,'}, {'start': '541.760', 'end': '550.000', 'text': 'one x will be left in the denominator'}, {'start': '543.480', 'end': '554.640', 'text': 'plus a / x² = 0 Now you have to'}, {'start': '550.000', 'end': '558.440', 'text': 'keep in mind here that if I common a from here'}, {'start': '554.640', 'end': '563.519', 'text': 'then it will come to me'}, {'start': '558.440', 'end': '567.200', 'text': 'the square of x² + 1 / x. Then'}, {'start': '563.519', 'end': '573.680', 'text': 'if I common b from here then it will come to me'}, {'start': '567.200', 'end': '576.399', 'text': 'x + 1 / x + c = 0 Now the'}, {'start': '573.680', 'end': '583.080', 'text': 'substitution that we have to let out in this will be that'}, {'start': '576.399', 'end': '585.920', 'text': 'if you let x + 1 / x ='}, {'start': '583.080', 'end': '587.839', 'text': 'y, we will square both the sides.  I have'}, {'start': '585.920', 'end': '590.839', 'text': 'less space, so I have started doing this step here'}, {'start': '587.839', 'end': '593.080', 'text': 'that x + 1 /'}, {'start': '590.839', 'end': '596.240', 'text': 'x² ='}, {'start': '593.080', 'end': '600.839', 'text': 'y² Now if you open this formula here'}, {'start': '596.240', 'end': '606.440', 'text': 'then it will come a² + 2'}, {'start': '600.839', 'end': '610.399', 'text': 'a b + b² ='}, {'start': '606.440', 'end': '612.720', 'text': 'y² This x will cancel with this x.  So'}, {'start': '610.399', 'end': '619.519', 'text': 'this plus two will shift there.  That means you will get'}, {'start': '612.720', 'end': '623.440', 'text': 'x² + 1 / x² = y² and this +2 will'}, {'start': '619.519', 'end': '626.079', 'text': 'go there and get -2.  So when I put this value'}, {'start': '623.440', 'end': '628.720', 'text': 'back in my substitution, the'}, {'start': '626.079', 'end': '632.519', 'text': 'further equation that will be formed for you will be'}, {'start': '628.720', 'end': '632.519', 'text': 'in this form.'}, {'start': '633.680', 'end': '641.880', 'text': 'You can see that'}, {'start': '636.959', 'end': '644.760', 'text': 'instead of a x² +1 / x² you will get y² -'}, {'start': '641.880', 'end': '646.360', 'text': '2 y²'}, {'start': '644.760', 'end': '653.480', 'text': '-'}, {'start': '646.360', 'end': '659.440', 'text': '2 + b instead of x + 1 / x you will get'}, {'start': '653.480', 'end': '662.839', 'text': 'y plus c = 0 so this will come to you the'}, {'start': '659.440', 'end': '670.000', 'text': 'square of y minus minus'}, {'start': '662.839', 'end': '672.839', 'text': '2a + by + c = 0 so if we'}, {'start': '670.000', 'end': '677.120', 'text': 'try to make it in standard form first the square of y,'}, {'start': '672.839', 'end': '683.200', 'text': 'then y and then the constant will come'}, {'start': '677.120', 'end': '686.480', 'text': 'c - 2a = 0 so this will come to us in'}, {'start': '683.200', 'end': '686.480', 'text': 'reduced quadratic'}, {'start': '688.440', 'end': '692.480', 'text': 'form, these are its steps.  Let me repeat,'}, {'start': '691.040', 'end': '694.880', 'text': 'if we have an'}, {'start': '692.480', 'end': '697.040', 'text': 'equation such that if we replace x'}, {'start': '694.880', 'end': '699.600', 'text': 'with 1/x and it makes no difference, then'}, {'start': '697.040', 'end': '701.920', 'text': 'you have a reciprocal equation.  The'}, {'start': '699.600', 'end': '704.000', 'text': 'first step you will have to do is to'}, {'start': '701.920', 'end': '706.880', 'text': 'divide x² by all of your factors.'}, {'start': '704.000', 'end': '709.440', 'text': 'Then, factor out the terms containing x².'}, {'start': '709.440', 'end': '713.839', 'text': 'Separate the x and reciprocal terms, and isolate the constant.'}, {'start': '711.680', 'end': '716.240', 'text': 'Then we will apply this substitution method on it.'}, {'start': '713.839', 'end': '719.440', 'text': 'Now let us solve one example of this.'}, {'start': '716.240', 'end': '722.079', 'text': 'You have got the example.'}, {'start': '719.440', 'end': '726.240', 'text': 'So the first step we said was that'}, {'start': '722.079', 'end': '730.519', 'text': 'we would divide by x².'}, {'start': '726.240', 'end': '737.959', 'text': 'Divide x² on both sides.  So this will come to'}, {'start': '730.519', 'end': '737.959', 'text': '6x4 divided by x² - 35x'}, {'start': '738.000', 'end': '741.560', 'text': '/'}, {'start': '738.680', 'end': '745.959', 'text': 'x² +'}, {'start': '741.560', 'end': '752.959', 'text': '62x² divided by x²'}, {'start': '745.959', 'end': '755.200', 'text': '35x / x² + 6 divided by x² = 0 Here,'}, {'start': '752.959', 'end': '757.079', 'text': 'when we cut it, you will get'}, {'start': '757.079', 'end': '765.040', 'text': '6x² -'}, {'start': '759.560', 'end': '768.120', 'text': '35x This will cancel with 62 This'}, {'start': '765.040', 'end': '775.760', 'text': 'will cancel with one, so one will be left here.'}, {'start': '768.120', 'end': '778.079', 'text': '35 / x + 6 / x² = 0 First we have to collect x² and'}, {'start': '775.760', 'end': '781.760', 'text': 'its reciprocal.  If'}, {'start': '778.079', 'end': '783.639', 'text': 'I common out the six from here,'}, {'start': '781.760', 'end': '789.519', 'text': 'you get'}, {'start': '783.639', 'end': '793.200', 'text': 'x² + 1 / square of x.  Then in the next step,'}, {'start': '789.519', 'end': '795.079', 'text': 'if I make -35 common from here,'}, {'start': '793.200', 'end': '800.399', 'text': 'it will become'}, {'start': '795.079', 'end': '805.680', 'text': '-35. This will become x + 1 /'}, {'start': '800.399', 'end': '809.440', 'text': 'x + 62 = 0. We will name this as'}, {'start': '805.680', 'end': '814.000', 'text': 'equation number one.  So first of all we will make a'}, {'start': '809.440', 'end': '818.560', 'text': 'substitution here that let'}, {'start': '814.000', 'end': '822.320', 'text': 'x + 1 / x ='}, {'start': '818.560', 'end': '828.560', 'text': 'y on both sides then it will come that'}, {'start': '822.320', 'end': '831.800', 'text': "square of x + 1 / x = square of y, let's"}, {'start': '828.560', 'end': '835.480', 'text': 'solve it fundamentally.  Open the formula.'}, {'start': '831.800', 'end': '839.480', 'text': 'a² + 2'}, {'start': '835.480', 'end': '841.800', 'text': 'a b +'}, {'start': '839.480', 'end': '847.680', 'text': 'b² ='}, {'start': '841.800', 'end': '850.639', 'text': 'y² This x will cancel out with this x.  x² Do this 1'}, {'start': '847.680', 'end': '853.760', 'text': '/ x² here.'}, {'start': '850.639', 'end': '857.600', 'text': 'y² And this one which is +2, when it goes there'}, {'start': '853.760', 'end': '860.720', 'text': 'it will come to you -2.  So'}, {'start': '857.600', 'end': '864.399', 'text': 'we put this value into our equation one.  So'}, {'start': '860.720', 'end': '864.399', 'text': 'now you can say that one'}, {'start': '865.079', 'end': '875.600', 'text': 'becomes this, you will get'}, {'start': '868.920', 'end': '881.680', 'text': '6, instead of x² + 1 / x² we will put'}, {'start': '875.600', 'end': '885.680', 'text': 'y square minus 2 - 35,'}, {'start': '881.680', 'end': '886.880', 'text': 'instead of x +1 / x you will get'}, {'start': '885.680', 'end': '892.199', 'text': 'y'}, {'start': '886.880', 'end': '898.079', 'text': '+ 62 = 0, multiply it.'}, {'start': '892.199', 'end': '902.959', 'text': '6y² 6 * 2 12 - 35'}, {'start': '898.079', 'end': '905.240', 'text': 'y + 62 = 0 The square will come first in the standard form.'}, {'start': '907.880', 'end': '917.079', 'text': 'In 6y² - 35y and 62, when we minus 12'}, {'start': '912.000', 'end': '924.000', 'text': 'we get 62 +'}, {'start': '917.079', 'end': '924.000', 'text': '50 = 0 so we get the'}, {'start': '925.160', 'end': '931.440', 'text': 'reduced'}, {'start': '928.120', 'end': '935.199', 'text': "quadratic form.  Let's move towards it next"}, {'start': '931.440', 'end': '938.079', 'text': 'time.  Type Number Four'}, {'start': '935.199', 'end': '941.680', 'text': 'Exponential Equation An equation'}, {'start': '938.079', 'end': '945.120', 'text': 'involving an expression of the form k to the power'}, {'start': '941.680', 'end': '950.480', 'text': 'x is called an exponential equation.'}, {'start': '945.120', 'end': '954.000', 'text': 'Its general time you have that a kx + b k'}, {'start': '950.480', 'end': '958.680', 'text': 'x + c = 0 In this, the substitution that we make is that'}, {'start': '954.000', 'end': '961.680', 'text': 'let k power x ='}, {'start': '958.680', 'end': '967.920', 'text': 'y then you will have squaring on'}, {'start': '961.680', 'end': '970.320', 'text': 'both sides, square of k power x = y² If we'}, {'start': '967.920', 'end': '975.360', 'text': 'multiply these with each other then you will get'}, {'start': '970.320', 'end': '977.360', 'text': 'k power 2x = square of y'}, {'start': '975.360', 'end': '982.000', 'text': 'If we put this substitution in our main equation'}, {'start': '977.360', 'end': '985.560', 'text': 'then you will get a k'}, {'start': '982.000', 'end': '993.680', 'text': 'power 2x'}, {'start': '985.560', 'end': '998.440', 'text': 'y² + b k power x You will get y + c ='}, {'start': '993.680', 'end': '1003.240', 'text': '0 then you will get'}, {'start': '998.440', 'end': '1003.240', 'text': 'reduced quadratic form. Let us'}, {'start': '1004.800', 'end': '1010.680', 'text': 'discuss one example of this.  So'}, {'start': '1007.920', 'end': '1017.680', 'text': 'firstly we will make the substitution that'}, {'start': '1010.680', 'end': '1022.199', 'text': 'let 2 to the power x = y then next step will be'}, {'start': '1017.680', 'end': '1022.199', 'text': 'squaring on both sides.   The'}, {'start': '1027.439', 'end': '1038.600', 'text': 'square of 2 to the power of x ='}, {'start': '1031.720', 'end': '1041.360', 'text': 'y² So you will get 2 to the power of 2x ='}, {'start': '1038.600', 'end': '1043.199', 'text': 'y² This substitution that we are making,'}, {'start': '1041.360', 'end': '1048.319', 'text': 'if I put it in our main equation,'}, {'start': '1043.199', 'end': '1050.520', 'text': 'it will give you 2^2x It will come to'}, {'start': '1050.520', 'end': '1063.520', 'text': 'y² - 2 2 to the power of x It will come to you y +'}, {'start': '1058.320', 'end': '1066.160', 'text': '1 = 0 So you will get the reduced'}, {'start': '1063.520', 'end': '1066.160', 'text': 'quadratic'}, {'start': '1071.400', 'end': '1079.559', 'text': "form.  So let's move towards its type five."}, {'start': '1074.880', 'end': '1089.280', 'text': 'Type number five.  Equation of the'}, {'start': '1079.559', 'end': '1091.440', 'text': 'form x + a * x + b * x + c * x + d = k'}, {'start': '1089.280', 'end': '1093.760', 'text': 'Now what you have to do in this is'}, {'start': '1091.440', 'end': '1096.160', 'text': 'product the two pairs in such a way'}, {'start': '1093.760', 'end': '1098.480', 'text': 'that when we solve it, there should be'}, {'start': '1096.160', 'end': '1101.280', 'text': 'some term which is common in both.'}, {'start': '1098.480', 'end': '1105.200', 'text': 'It is not necessary that you'}, {'start': '1101.280', 'end': '1109.120', 'text': 'can have a + b always equal to c + d.'}, {'start': '1105.200', 'end': '1112.320', 'text': 'You can also have such that a + c = b'}, {'start': '1109.120', 'end': '1114.320', 'text': '+ d We can set law of addition or law of'}, {'start': '1112.320', 'end': '1116.640', 'text': 'multiplication according to our'}, {'start': '1114.320', 'end': '1118.400', 'text': 'criteria.  For'}, {'start': '1116.640', 'end': '1123.280', 'text': 'example, we have'}, {'start': '1118.400', 'end': '1126.960', 'text': 'taken this example that x + 1, x + 2, x + 3'}, {'start': '1123.280', 'end': '1130.520', 'text': 'and x + 4. If you see, I can say that'}, {'start': '1126.960', 'end': '1136.720', 'text': 'by the law'}, {'start': '1130.520', 'end': '1140.640', 'text': 'of addition, if I do it like this that 2 + 3 = 1'}, {'start': '1136.720', 'end': '1142.880', 'text': '+ 4, in this case, these two'}, {'start': '1140.640', 'end': '1145.120', 'text': 'can come to you as equal to five.  This means that'}, {'start': '1142.880', 'end': '1147.600', 'text': 'I will multiply the pairs with each other'}, {'start': '1145.120', 'end': '1151.360', 'text': 'in this scenario that I'}, {'start': '1147.600', 'end': '1154.400', 'text': 'will multiply the first with the last and then I'}, {'start': '1151.360', 'end': '1156.080', 'text': 'will multiply the second with the third pair.  So'}, {'start': '1154.400', 'end': '1161.280', 'text': 'I have started applying the same thing here that it will'}, {'start': '1156.080', 'end': '1164.640', 'text': 'come x + 1 * x + 4, I will multiply these two pairs with each other'}, {'start': '1161.280', 'end': '1170.480', 'text': 'and then you will get'}, {'start': '1164.640', 'end': '1174.000', 'text': 'x + 2 and x + 3, I will multiply them with each other.'}, {'start': '1170.480', 'end': '1177.919', 'text': 'Now'}, {'start': '1174.000', 'end': '1180.600', 'text': 'if we multiply this first pair with each other.   We'}, {'start': '1177.919', 'end': '1183.840', 'text': 'will multiply x by this.  We will get'}, {'start': '1180.600', 'end': '1187.919', 'text': 'x² If we multiply this x by this then we will get'}, {'start': '1183.840', 'end': '1193.360', 'text': '4x By multiplying it by one, we will get the same'}, {'start': '1187.919', 'end': '1198.720', 'text': 'term again x + 4 Now you will get'}, {'start': '1193.360', 'end': '1201.840', 'text': 'x² 4x + x This will get you 5x'}, {'start': '1198.720', 'end': '1204.240', 'text': '+ 4 We have solved the first equation, the'}, {'start': '1201.840', 'end': '1207.760', 'text': 'first part If we solve the second part of this, then'}, {'start': '1204.240', 'end': '1209.520', 'text': 'you will get this If we'}, {'start': '1207.760', 'end': '1212.720', 'text': 'multiply this x by this, we will get the'}, {'start': '1209.520', 'end': '1217.280', 'text': 'square of x'}, {'start': '1212.720', 'end': '1222.880', 'text': 'Multiply this x by three will get 3x'}, {'start': '1217.280', 'end': '1225.919', 'text': 'Multiply this by 2x and we will get 3 * 2'}, {'start': '1222.880', 'end': '1230.960', 'text': '6 And if we solve this further the'}, {'start': '1225.919', 'end': '1235.480', 'text': 'square of x 3x + 2x will get 5x + 6 And'}, {'start': '1230.960', 'end': '1238.720', 'text': 'you will get equal to'}, {'start': '1235.480', 'end': '1240.960', 'text': '24 So now if you see the common thing in both of these parts, then'}, {'start': '1238.720', 'end': '1245.200', 'text': 'this value is'}, {'start': '1240.960', 'end': '1251.679', 'text': 'common to both of them x² + 5x So we'}, {'start': '1245.200', 'end': '1253.440', 'text': 'will let x² + 5x = y When'}, {'start': '1251.679', 'end': '1257.880', 'text': 'we put this value in our substitution, then'}, {'start': '1253.440', 'end': '1263.320', 'text': 'you will get this  y +'}, {'start': '1257.880', 'end': '1266.480', 'text': '4 Next will come y + 6 ='}, {'start': '1263.320', 'end': '1269.840', 'text': '24 Now multiply these values \u200b\u200bfurther.  If we'}, {'start': '1266.480', 'end': '1273.360', 'text': 'multiply this y by this, we will get the'}, {'start': '1269.840', 'end': '1277.600', 'text': 'square of y. If we multiply this y by this, we will get'}, {'start': '1273.360', 'end': '1281.799', 'text': '6y. If we'}, {'start': '1277.600', 'end': '1286.880', 'text': 'multiply this by this, we will get 4y and 4 *'}, {'start': '1281.799', 'end': '1290.360', 'text': '6 24 is equal to 24. Solve this a little further and'}, {'start': '1286.880', 'end': '1297.440', 'text': 'you will get the square of y,'}, {'start': '1290.360', 'end': '1303.039', 'text': '6y + 4y. This will come to 10y plus 24'}, {'start': '1297.440', 'end': '1305.320', 'text': 'and this + 24 goes over there and - 24 = 0. This will cancel out with this.'}, {'start': '1305.320', 'end': '1317.720', 'text': 'y² + 10y + 0 = 0 This will give you'}, {'start': '1313.120', 'end': '1317.720', 'text': 'reducible quadratic form.'}, {'start': '1324.080', 'end': '1328.640', 'text': 'Here we were just solving the equations.'}, {'start': '1326.559', 'end': '1330.559', 'text': 'All we were doing was that if'}, {'start': '1328.640', 'end': '1332.960', 'text': 'you have an equation which is'}, {'start': '1330.559', 'end': '1335.280', 'text': 'not in quadratic form, then we'}, {'start': '1335.280', 'end': '1339.760', 'text': 'took it to quadratic form using one of these substitutions.'}, {'start': '1337.360', 'end': '1341.280', 'text': 'So these were some basic concepts.  There'}, {'start': '1339.760', 'end': '1343.520', 'text': 'were different types of equations.  We discussed all those'}, {'start': '1341.280', 'end': '1345.840', 'text': 'questions.  If you'}, {'start': '1343.520', 'end': '1347.679', 'text': 'want to visit my notes then'}, {'start': '1345.840', 'end': '1350.400', 'text': 'I will put its link in the comment section.  If you'}, {'start': '1347.679', 'end': '1352.799', 'text': 'want to visit the notes, you'}, {'start': '1350.400', 'end': '1356.799', 'text': 'can do so easily.  Finally a small'}, {'start': '1352.799', 'end': '1361.520', 'text': 'message for your viewers and listeners, do'}, {'start': '1356.799', 'end': '1365.360', 'text': 'good to people.  A favor is a'}, {'start': '1361.520', 'end': '1367.840', 'text': 'treasure that never ends.  Dear'}, {'start': '1365.360', 'end': '1369.760', 'text': 'children, take great care of yourself.  See you'}, {'start': '1367.840', 'end': '1372.000', 'text': 'in the next video.  Till then Allah'}, {'start': '1369.760', 'end': '1372.000', 'text': 'Hafiz.'}]