Solving a Quadratic Equation by Factorisation Method

Hinzugefügt: 2026-05-15

[00:00:07][music] Hello students, welcome to my YouTube channel Lealam.

[00:00:19]Today we are going to learn a very important concept in mathematics factorizing a quadratic polomial.

[00:00:29]Many students feel this topic is difficult.

[00:00:33]They often get confused and make mistakes while factorizing.

[00:00:39]But don't worry, in this video I am going to explain it in a very simple and clear way.

[00:00:49]Actually there are three methods to solve a quadratic equation.

[00:00:56]Number one, factorization method. Number two, quadratic formula method. Number three, completing square method. Out of these three methods, the factorization method is very very very important one.

[00:01:16]This concept is extremely important for students who are studying 10th class mathematics and it is also very useful in higher studies. Not only that, this factorization method helps us to solve quadratic equations easily.

[00:01:36]Nowadays, in many educational videos, teachers explain only shortcut methods.

[00:01:43]Those shortcuts may be useful in competitive examinations where speed is important. But in descriptive examinations, you will get full marks only when you solve the problem in a proper step-by-step procedure.

[00:02:02]So in this video, I will teach you the correct systematic method in an easy way. If you learn this method properly and practice many problems, then slowly you will be able to guess the factors very easily without confusion. Most students forget the steps or use the wrong method. So I will also share techniques to remember the process, avoid common mistakes and solve problems confidently. After watching this video carefully, factorization will become much easier for you. So let's go into the video.

[00:02:45]This factorization mainly depends on four important models of quadratic polomials.

[00:02:53]Let me explain each type clearly step by step.

[00:02:58]Let us take the first quadratic equation that is x² + 7 x + 12 is equal to zero. This is the first model of quadratic equation we have taken. First we need to split this quadratic polomial into two factors of linear polomials.

[00:03:32]For doing that we need to split this middle term into two terms.

[00:03:40]So in the process first we need to multiply the x² term with the constant term. We need to multiply this x² term with the constant term. So here the x² term is x² x x² into + 12 then we get + 12 x² that is + 12 x².

[00:04:12]My dear students, now we need to write the factors of this 12. Let us write the factors of the 12 that is 1 12 are 12 and we can write 2 6 are 12 also we can write 3 4 are 12.

[00:04:36]Here is the main important topic of this video that is since we got a plus symbol we have multiplied this x² term with the constant term that is we obtained + 12 x² here this plus indicates plus indicates addition we have to write here addition this plus indicates addition Naturally when do we do the addition? If both factors have plus symbol or both factors have minus symbol then we do the addition. So here for these factors we need to write either both plus symbol or both minus symbol that depends upon the middle term of the quadratic equation.

[00:05:35]Here in the middle term plus symbol is there that's why for these two factors we need to write both plus symbol. Let us write both factors with the plus symbol. This is plus plus this is also plus this is also plus.

[00:05:58]Now look at this factors by simplifying which factors we obtained this middle term that factors we need to select. So if you take + 1 and + 12 we get uh + 13 that is not middle term + 2 + 6 that is + 8 that is not the middle term and if you add + 3 + 4 we get + 7 that is why we need to select these two factors.

[00:06:30]These two factors to be selected to write instead of this middle term. This x² x² we need to write one x term here another x term here. That means this factors become + 3x + 4x those two we need to split this 7 x into the terms + 3x + 4x. Let us write instead of this middle term we will write these two terms. Then we can write here x² + 3x + 4x + 12 is = 0.

[00:07:17]Then in these two terms we need to take some term common and we need to take some term common in these two terms.

[00:07:30]Then see what is the HCF of these two numbers here x² means x into x here 3x that means we can take x common in both terms. Let us write x common in both terms. Then it becomes x of x + 3. And the hCF of 4x and 12 is 4. Means 4 can be divided with both terms. So obviously we will take four as common in these two terms. Then we get x + 4 is taken common in this 12. If you take four common we get three that is equal to zero. Then if you observe here x + 3 and here also x + 3 is common in that's why we can take x + 3 common again. If you take x + 3 common again we get x + 4.

[00:08:35]Here x is remaining here four is remaining. So we get x + 4 is equal to 0.

[00:08:43]Then if the product of these two linear polomials is equal to zero then each of the linear polomials may be equal to zero. That's why we can write x + 3 is =0 or we can write x + 4 is equal to z.

[00:09:05]Then if x + 3 is equal to 0, x becomes -3 or x + 4 is equal to 0, x becomes -4.

[00:09:16]So x = -3 and x = -4 are the roots of this quadratic equation. Like this we have to solve the quadratic equation.

[00:09:29]Model number one. Let us see the second model of this solutions. Let us take here x² - 14x + 24 is equal to zero.

[00:09:43]Then remember what is the first step of writing the factorization. We need to multiply x² term with the constant term.

[00:09:53]So x² term is x² constant term is + 24.

[00:09:58]If you multiply both we get + 24 x². Now we need to write the factors for this 24. The factors of 24 are 1 24s are 24.

[00:10:12]2 12 are 24. 3 8 are 24 and 4 6 are 24.

[00:10:18]These are the factors of this 24. Again we have to go look at this symbol. This symbol indicates again addition.

[00:10:28]Addition we need to write like this for habitating purpose. This plus indicates addition. When do we do addition?

[00:10:37]Addition means both positive symbol or both negative symbol. That means this coding indicates for these factors we need to write either both positive or both negative.

[00:10:53]that depends upon the middle term of the quadratic equation. In the middle term of the quadratic equation here minus symbol is there. That's why for this both factors we need to write minus symbols.

[00:11:12]Now understand the difference between this model and this model. Here we obtained + 12 x² plus means addition means both plus both minus that depends upon the middle term. Middle term is plus that means both factors we need to write plus symbol here again + 24 x² is there that indicates addition both plus or both minus again that depends upon the middle term minus here minus4 is there that's why we have to write both factors minus here both factors plus here both factors minus because here middle term is plus that's why both factors Plus here middle term is minus that's why both factors is minus we have to write. Now the important thing is selection of the factors. So by simplifying which pair of factors we obtained the middle term those factors we need to select. Then if you see here -1 -4 gives - 255 that is not the middle term -2 -12. If you simplify we get minus4 which is minus4 is the middle term. That's why we need to select these two middle terms.

[00:12:37]So here x² symbol is there that we have to split as x 1 x here another x here.

[00:12:44]So instead of this -4x now we need to split it as - 2x - 12x. Let us write the middle term as - 2x and -2x.

[00:12:58]Let us write this one. x² - 2x - 12x + 24 is equal to zero. Again in these two terms we need to take common of the gcd and we need to take the gcd of these two terms as common. So if you see x² on 2x the x we can take common as their gcd then we can write x -2 because x is taken common only two is remaining here. Then here and here we can take -12 common. Whenever you see this third term negative we need to take negative term compulsory as common. Then we take -12 as common. Then we get -12 of x is remaining here. In this plus we can write two negative symbols because minus into minus is plus. If you take 1 minus is common then we get another minus here.

[00:14:11]Then 12 table we can divide 24. 24 by 12 it is equal to 2. Then we get here 12 is equal to zero. Then see x - 2 is the first two terms common and x -2 is common in second two terms also. That's why again we need to take x -2 common. x -2 if you take common then we get uh x - 12 because here x -2 is taken common. X - 2 is taken common here X is remaining here 12 is remaining that is equal to zero if the product of these two terms is equal to zero again obviously X - 2 becomes zero or X - 12 becomes zero.

[00:15:05]Then if x - 12 is =0 x = 2 or x - 12 is equal to 0, x= 12. That means this 12, this 12 and 2 are the roots of the quadratic equation.

[00:15:27]Then you understand these two methods.

[00:15:30]Then see in the both models we obtained plus symbol here constant term has positive symbol that indicates addition.

[00:15:42]Here + 12 that means + 12 x². Here + 24 means + 24 x² that indicates addition.

[00:15:51]Both factors we need to add. Both factors we need to add. When do we do addition? We have to if you keep two positive symbols or two negative symbols then we will add that's why here middle term contains minus that's why we kept both factors minus here the middle term contains plus that's why both factors we kept positive I hope you understand this model number one and model number two let us move to the model number three and model number four let us Take another example in the model number three. x² + 5x - 36 = 0. As you know the first step in solving the quadratic equation, we need to multiply x² term with the constant term. There we get - 36 x². Because here the constant term contains negative symbol we will get - 36 x². So - 36 x² means it represents subtraction. When do we do the subtraction? If one symbol is plus another symbol is minus then we do the subtraction. This means the coming factors, pairs of factors, we need to write one positive symbol, one negative symbol. Let us write the factors of 36 first. The factors of 36 are 1 into 36, 2 into 18, 3 into 12, 4 into 9. These are the factors of 36.

[00:17:35]Then as we discussed earlier we need to write one positive2 symbol one negative symbol for these factors.

[00:17:45]Again that depends upon the middle term.

[00:17:49]See here the middle term is positive.

[00:17:52]That's why for the bigger number we need to write positive and the smaller number is negative. Bigger number is positive and the smaller number is negative.

[00:18:04]Bigger number positive, smaller number negative. Bigger number positive, smaller number negative. Like this we have to write.

[00:18:13]Then we need to simplify these terms. -1 + 36 which is equal to + 35 which is not the middle term. -2 + 18 which is equal to + 16 which is not the middle term. + 12 - 3 which is equal to + 9 which is not the middle term but + 9 and -4 which is equal to + 5 that's why we have to select these two factors these two factors we need to select obviously this x² will be there that x² we can write one x here another x is here that's why we need to split this + 5x as -4x + 9x.

[00:19:02]Let us do it. Then we can write this quadratic equation as x² - 4x + 9x - 36 is equal to zero. See here this is + 5x that is split into - 4x + 9x. Then we can take x common here. Take x common then we get x - 4. And in these two terms we can take + 9 common then we get x - 4 which is equal to zero. Then again here x - 4 and x - 4 will be there. So we can write x - 4 as common again x - 4 of x + 9 is = 0. Then look at dear students x - 4 will be equal to zero or x + 9 will be equal to zero. Then we get x = 4 or x = -9. Like this we can solve this quadratic equation model number three. Let us go into the model number four. Let us take this example. x² - 7 x - 60 is equal to 0.

[00:20:23]Let us move to the first step that x² term we need to multiply with the 60 that we get -60 x² we need to multiply with the negative symbol also x² into -60 becomes -60 x² this minus indicates subtraction. Subtraction means one positive symbol, one negative symbol.

[00:20:47]Let us write the factors of 60 that we can write 1 into 60, 2 into 30, 4 into 15, 5 into 12 and 6 into 10. These are the factors of 60.

[00:21:05]For the pairs of these factors, we need to write one positive symbol, one negative symbol. Which number positive?

[00:21:12]Which number negative? that is decided by middle term. Here the middle term contains negative symbol. That's why for the bigger number we have to write negative symbol and the smaller number should contain positive symbol. Bigger number minus small number positive.

[00:21:30]Bigger number minus small number positive. Bigger number minus small number plus minus plus. like this we have to separate the signs of the pairs of these factors. Then if you simplify these factors for which factors we get this middle term those factors we need to be selected. So here + 1 - 60 g use - 59 which is not middle term + 2 -30 which is also not middle term + 4 - 15 that is -1 which is not equal to - 7 + 5 - 12 will be equal to -7 that's why we need to select these two term. We can split this middle term into + 5 and -12.

[00:22:20]So obviously this x² will be there that x² x² we can write 1 x here another x is here then we can write we can split this - 7x as + 5x - 12x let us do it then we can write x² - 12x - 60 is = 0.

[00:22:45]Then dear students in these two terms we can take x common that is x + 5. In these two terms we can take -12 common then we get x - x + 5 again.

[00:23:02]Then if you take x + 5 common again then we get x -2 is = 0. Then x + 5 =0 or x -12 is =0. Then x = -5 or x = -12. These are the roots of the quadratic equation.

[00:23:29]You understand? In this model number three and model number four, it is mainly depended with this product. x² into - 36 is equal to - 36 x². Here -60 x²us indicates subtraction. Subtraction means one positive symbol, another one is negative symbol. That depends again on the middle term. If the middle term contains plus, then we need to write for bigger number plus smaller number minus.

[00:24:01]If the middle term contains negative symbol, for bigger number we need to write negative symbol and smaller number we need to write positive symbol. Like that we need to split the middle term into two factors.

[00:24:18]Let me conclude this video with another model. We will take 2x² - 15x - 45 is equal to 0. Then as we discussed in the early problems then here the x² term should be multiplied with the constant term. Here the x² term is not x² like previous problems but it contains 2 as the coefficient. So we need to include this coefficient also while multiplying with the constant term. So we get 2x² into - 455 which gives -90 x² 2x² - 45 we need to multiply both then we get to -9 x² then for this 90 we need to write the factors not for 45. This is the main thing we have to understand. We need to multiply 2x² with - 45 and we get the product as - 90 x². We need to write the factors for 90 not for 45. Some students they will do by seeing the constant term and they will write the factors for the only constant term. But it is wrong technique. You need to multiply the coefficient of x² also with the constant term. Then we get - 90 x². Remaining procedure is the same as we learned in the previous four models. Let us write the factors for 90. Then we can write 1 into 90 and we can write 2 into 45. 3 into 30. Five table we can write 5 into 80. 6 table we can write 6 15s are 90.

[00:26:07]Then this minus symbol as we learned it indicates subtraction. It means 1 + and 1 minus that depends upon the middle term. Middle term is negative here.

[00:26:21]That's why we need to write for bigger number minus and small number positive.

[00:26:27]Bigger number minus small number plus minus plus plus min - plus plus minus and this is plus. If you simplify in this factors if you simplify + -8 we get -5.

[00:26:47]That's why we need to select these two factors. This x is there. That x we can write we can write 1 x here and another x is here. Then let us split this -5x as + 5x - 18x. Then we get 2x² + 5x - 18x - 45 is equal to 0. Then 2x² + 5x we can take only x as common. Take x as common. Then we get here 2x + 5 and here and here we can take - 9 as common - 9 if you take common 9 2's are 18 and x will be there minus is taken common then we get a positive symbol here 9 5 are 45 then we can write five is equal to zero then obviously this bracket is 2x + 5 this bracket also 2x + 5 then let us Take 2x + 5 as common. If you take 2x + 5 common then x - 9 will be remaining.

[00:27:55]And uh if you take this 2x + 5 is equal to zero. Then 2x becomes -5 x becomes - 5x2 or x - 9 is =0 x becomes 9. So x= 5x2 and x = 9 are the the roots of this quadratic equation.

[00:28:24]So this is the method of factorizing a quadratic polomial or a quadratic equation and to get the roots of the quadratic equations.

[00:28:36]My dear students, if you practice many problems in this model, then definitely you will get perfection and you will get the full out of full marks in the examinations.

[00:28:50]I hope you understand and you like the video. If you like the video, please subscribe my channel. Thank you very much for watching this video.