Don't Think About LCM Immediately..... Rather Do This!

Added: 2026-05-05

[00:00:02]Hello friends, welcome back to MC Gyan.

[00:00:05]Today in this video we have one very very interesting and challenging question from rational equations, which will be solving for the real values of X.

[00:00:16]So, let's get it started by writing here denominators cannot be zero. So, X cannot be equal to 4 5 7 8.

[00:00:30]Now, we are going to check the square of denominators.

[00:00:34]Here we have X minus 4.

[00:00:38]Its a square will be X is square minus 8X plus 16. Here we have 14.

[00:00:47]If you will check a square of X minus 8, then it is X is square minus 16X plus 64. Here we have 62.

[00:00:56]The square of X minus 5 is X is square minus 30X. Here we have 26.

[00:01:02]But as per formula it is 25.

[00:01:05]And X minus 7 whole is square will be X is square minus 14X plus 49. Here we have 50.

[00:01:13]So, let's write our numerator in terms of denominator is square.

[00:01:19]I will write the equation X is square minus 8X will write 14 as 16 minus 2.

[00:01:30]In denominator will write X minus 4.

[00:01:34]Similarly here will write X is square minus 16X plus 64 minus 2 over X minus 8.

[00:01:49]equal to in RHS will write X is square minus 10X plus 25 plus 1 over X minus 5 plus X is square minus 14X plus 49 plus 1.

[00:02:15]And in denominator will write X minus 7.

[00:02:19]Now, we are going to pick first three terms of numerators.

[00:02:25]These are perfect squares.

[00:02:30]Let's write perfect is square of their respective denominators. We will write here X minus 4 whole is square minus 2 over X minus 4 plus X minus 8 whole is square minus 2 over X minus 8 equal to X minus 5 whole is square plus 1 over X minus 5 plus X minus 7 whole is square plus 1 over X minus 7.

[00:03:17]Now, we'll split numerator with respect to their denominators.

[00:03:22]So, we can write X minus 4 whole is square over X minus 4 X minus 4.

[00:03:28]Then we have minus 2 over X minus 4 plus X minus 8 whole is square over X minus 8 is X minus 8 minus 2 over X minus 8 equal to X minus 5 plus 1 over X minus 5 plus X minus 7 plus 1 over X minus 7.

[00:04:01]Now, we will add X minus 4 and X minus 8 from LHS.

[00:04:10]We can write X plus X is 2X.

[00:04:13]minus 4 minus 8 is minus 12.

[00:04:16]From RHS we have X minus 5 and X minus 7 which will give us 2X minus 12 after adding them.

[00:04:28]Now, remaining terms would be as it is.

[00:04:31]We'll write minus 2 over X minus 4 minus 2 over X minus 8.

[00:04:39]In RHS we'll write plus 1 over X minus 5 plus 1 over X minus 7.

[00:04:50]Now, we'll subtract 2X minus 12 from both the sides.

[00:04:55]So, 2X minus 12 will get over.

[00:05:01]We will write our remaining equation minus 2 over X minus 4 minus 2 over X minus 8 equal to 1 over X minus 5 plus 1 over X minus 7.

[00:05:24]Now, we'll write all the terms to right hand side.

[00:05:28]So, we can write here 2 over X minus 4 plus 2 over X minus 8 plus 1 over X minus 5 plus 1 over X minus 7 equal to zero.

[00:05:50]We can take two common from first two terms and we'll be writing 1 over X minus 4 plus 1 over X minus 8 plus 1 over X minus 5 plus 1 over X minus 7 equal to zero.

[00:06:15]Now, we'll use substitution.

[00:06:18]Let us say X minus 6 this is equal to U.

[00:06:24]So, we'll write here X minus 4 in terms of U will be U plus 2.

[00:06:32]And if you will write X minus 8 then we have to write U minus 2.

[00:06:38]Similarly, we'll write X minus 5 will be equal to U plus 1.

[00:06:45]And X minus 7 will be U minus 1.

[00:06:51]So, our equation will become two times 1 over U plus 2 plus 1 over U minus 2 plus 1 over U plus 1 plus 1 over U minus 1 equal to zero.

[00:07:17]Now, we are going to take LCM. LCM of these two brackets.

[00:07:23]So, we will write here two times 1 over U plus 2 plus 1 over U minus 2 will write U plus 2 times U minus 2 in the denominator and we have to add in the numerator. So, U plus 2 plus U minus 2 plus in denominator we'll multiply both the denominator brackets U plus 1 times U minus 1 and in numerator we will add both U plus 1 plus U minus 1 equal to zero.

[00:08:07]Now, we'll cancel plus 2 and minus 2 plus 1 and minus 1.

[00:08:14]So, we will write two times in bracket we'll write 2U over A plus B times A minus We know that this is difference of two is square formula.

[00:08:27]A is square minus B is square. So, we'll write here U is square minus 2 is square plus U plus U will be 2U.

[00:08:39]In denominator we'll write U is square minus 1 is square equal to zero.

[00:08:48]But here we'll write bracket only because of multiplier is two with only first term.

[00:08:57]Now we will take 2U common from the numerator.

[00:09:02]So, I will take two inside and we will take 2U outside. So, we will write 2U in bracket we'll write 2 over U is square minus 4 plus 1 over U is square minus 1 equal to zero.

[00:09:26]Now, we'll use product zero rule.

[00:09:29]So, we can write either U equal to zero or 2 over U is square minus 4 plus 1 over U is square minus 1 equal to zero.

[00:09:46]From U equal to zero equation we have U value X minus 6. So, we will write here X minus 6 equal to zero.

[00:09:57]So X will be equal to six. Our first real solution is X equal to six.

[00:10:04]Now we have to solve this equation.

[00:10:08]We'll take LCM once again.

[00:10:11]So we can write Let me write equation here.

[00:10:15]Two over U is square minus four plus one over U is square minus one equal to zero.

[00:10:26]We'll multiply both the denominators in the denominator here. U is square minus four times U is square minus one.

[00:10:36]And in numerator we will multiply two with U is square minus one.

[00:10:42]And one with U is square minus four equal to zero.

[00:10:49]So denominator cannot be zero. We'll write here two times U is square minus one.

[00:10:55]Plus U is square minus four will be equal to zero. Or we can write two U is square minus two plus U is square minus four equal to zero.

[00:11:12]We'll add two U is square with U is square three U is square minus two minus four minus six equal to zero.

[00:11:22]We'll add six to both sides and write three U is square equal to six.

[00:11:29]Now we have to divide by three both sides.

[00:11:33]Three over three will be one.

[00:11:36]And six over three will be two.

[00:11:38]So we can write U is square equal to two.

[00:11:43]After taking a square root both sides, we'll write U will be equal to plus minus the square root of Now U is X minus six.

[00:11:55]So we will write U equal to plus minus the square root of two.

[00:12:03]In place of U we'll write X minus six equal to plus minus the square root two.

[00:12:10]After adding six to both sides, we are going to write here two solutions. X equal to six plus minus the square root.

[00:12:21]So in total we have three real solutions.

[00:12:25]X1 it is six.

[00:12:28]X2 it is six plus root two.

[00:12:33]And X3 will be equal to six minus root I hope friends you will like this video.

[00:12:41]Thank you so very much for watching. Do not forget to like, share, subscribe.

[00:12:45]Bye-bye till next video. Good luck.

[00:12:47]Bye-bye.