An Outstanding Irrational Equation | Can You Solve This?

[00:00:02]Welcome to InfyGyan, my dear friends. We have one very interesting irrational equation. It looks complicated, but a clever substitution makes the solution surprisingly simple.

[00:00:15]So, let's begin.

[00:00:16]I will take this whole bracket to the denominator of 10.

[00:00:23]So, I can write equation square root x + 6 + square root x + 8 = 10 over square root x + 20 + square root x + 10.

[00:00:47]Now, we will rationalize our denominator.

[00:00:50]So, we need to multiply and divide by the denominator's conjugate square root x + 20 square root x + 10.

[00:01:04]In the numerator also, square root x + 20 - square root x + 10.

[00:01:15]Now, we have a + b * a - b form in the denominator.

[00:01:21]So, we will use difference of two squares identity.

[00:01:26]So, our numerator will become 10 * square root x + 20 - square root x + 10.

[00:01:34]And in denominator we'll use difference formula and we'll write square root x + 20 whole square square root x + 10 whole square.

[00:01:50]Now, a square root and a square will get canceled out from the denominator.

[00:01:56]We will get 10 times the square root x + 20 minus the square root x + 10 over x + 20 minus of x + 10. So, minus x minus 10.

[00:02:19]Plus x minus x will over.

[00:02:22]We will cancel numerator 10 with the denominator 10 because denominator is 20 minus 10. So, I can cancel numerator also and denominator will become one.

[00:02:34]So, we'll write here the square root x + 20 minus the square root x + 10.

[00:02:45]And in LHS, we have the square root x + 6 plus the square root x + 8.

[00:02:52]I will take this negative term to the left-hand side.

[00:02:58]So, we can write our equation the square root x + 6 plus the square root x + 8 plus the square root x + 10 equal to the square root x + 20.

[00:03:22]Now, we will use substitution. Let's consider x + 8 suppose equal to u.

[00:03:32]So, x + 6 will become u minus 2.

[00:03:38]And x + 10 will become u + 2.

[00:03:44]In center, we'll write the square root u.

[00:03:49]In right-hand side, we have to write the square root U + 12.

[00:03:56]Now, we'll take the square root U to the RHS.

[00:04:01]So, we have the square root U minus 2 plus the square root U plus 2 equal to the square root U plus 12 minus the square root U.

[00:04:20]Now, we have to square both sides.

[00:04:24]We'll put power 2 in LHS and in RHS.

[00:04:30]Now, we'll be using A + B whole square and A - B whole square identities.

[00:04:37]So, I will get the square of the square root U minus 2, U minus 2 plus the square of the square root U plus 2, U plus 2 plus two times the square root of U minus 2 times U plus 2. We will use difference of two squares formula and we can write U square minus 2 square, which is 4 equal to the square of the square root U plus 12, U plus 12 the square of the square root U, U minus two times the square root of U times U plus 12.

[00:05:20]Now, we will subtract 2U from both the sides.

[00:05:26]So, 2U will be over from both the sides.

[00:05:29]Plus 2 minus 2 will get over.

[00:05:32]In LHS, we'll write two times the square root U square minus 4 and in RHS, there will be 12 minus two times the square root of U times u plus 12.

[00:05:50]Now, our equation is divisible by two.

[00:05:53]So, let us divide LHS by two and RHS by two.

[00:06:00]Two over two will become one.

[00:06:02]12 over two will become six. Two over two will become one.

[00:06:07]So, we will get a square root u square minus four equal to six minus a square root u times u plus 12.

[00:06:23]Now, we will rearrange the terms.

[00:06:27]So, we can write a square root u times u plus 12 in left-hand side.

[00:06:35]And in right-hand side, there will be six minus a square root of u square minus four.

[00:06:44]We have a square root in LHS.

[00:06:47]So, we'll be squaring both sides. We'll put power two in LHS and RHS.

[00:06:53]A square root will be over with a square in left-hand side. So, there will be u times u plus 12 in LHS.

[00:07:05]In RHS, we'll use a minus b whole square identity. Six square is 36 plus a square root a square is a square root free term u square minus four minus two times six 12 times a square root u square minus four.

[00:07:28]We'll write here u square plus 12 u equal to u square 36 minus four 32 minus 12 times a square root u squared minus four.

[00:07:47]Now, we will subtract u squared from both the sides.

[00:07:52]So, we'll be writing here 12 u in LHS.

[00:07:57]In RHS, there will be 32 minus 12 times the square root u squared minus four.

[00:08:07]Now, our equation is divisible by four.

[00:08:10]So, let's divide LHS by four and RHS by four.

[00:08:16]12 over four will be three.

[00:08:19]32 over four will be eight.

[00:08:22]12 over four will be three.

[00:08:25]So, we can write three u equal to eight minus three times the square root u squared minus four.

[00:08:37]Now, we'll take this negative term to the LHS and three u to the RHS.

[00:08:44]So, we will get three times the square root u squared minus four in left side and in right side, there will be eight minus three u.

[00:08:58]Now, we have only one is square root in LHS. So, we have to consider a squaring for the last time.

[00:09:06]We will get eight minus three u whole squared.

[00:09:12]It is a square of right hand side, which I am writing in left hand side.

[00:09:16]And in RHS, we will write our LHS, which is three times the square root u squared minus four whole squared.

[00:09:27]Now, we have to use a minus b whole squared formula here.

[00:09:32]We will get eight squared is 64 plus three u whole squared is nine u squared.

[00:09:40]2 * 8 * 3 48u = 3 ^ will be 9 u ^ - 4 which can be written as 9u ^ - 36 In LHS we'll write 9u ^ - 48u + 64 Now we will subtract 9u ^ from both the sides.

[00:10:13]Now we'll take -36 to the LHS and -48u to the RHS So there will be 64 + 36 in LHS and 48u in right hand side.

[00:10:31]100 will be equal to 48u We can divide our equation by 4. So let's divide both sides by 4.

[00:10:43]We will get 25 in left hand side and we will get 12u in RHS. So I can write 12u = 25 Now we have to divide by 12 to find the value of u.

[00:11:03]So 12 over 12 will become 1. u will be equal to 25 over 12.

[00:11:11]u will be 25 over 12 Now u was our substitution x + 8.

[00:11:20]So in place of u we can write x + 8 = 25 over 12 Now we'll subtract 8 from both the sides.

[00:11:33]So, plus eight minus eight will get over.

[00:11:36]We will write X will be equal to 25 over 12 minus eight can be written as with denominator 12 96.

[00:11:50]Now, in numerator there will be 25 minus 96 over 12.

[00:11:57]Which will give us our answer minus 71 over 12.

[00:12:04]Now, we have to check whether it is true as per our domain or not.

[00:12:09]So, we will find domain of the equation.

[00:12:14]So, I can write our equation is square root X plus six plus square root X plus eight plus square root X plus 10 equal to square root X plus 20.

[00:12:34]Radicands must be non-negative, greater than or equal to zero.

[00:12:39]Greater than or equal to zero.

[00:12:41]Greater than or equal to zero.

[00:12:44]Greater than or equal to zero.

[00:12:48]So, from first inequality we will get X should be greater than or equal to minus six.

[00:12:53]X should be greater than or equal to minus eight.

[00:12:57]X should be greater than or equal to minus 10.

[00:13:01]X should be greater than or equal to minus 20.

[00:13:05]And if we will take intersection of all the inequalities, we are going to get X should be greater than or equal to negative six.

[00:13:15]And our answer is minus 71 over 12, which is true.

[00:13:22]So, we will accept as per our domain.

[00:13:25]Our only real solution is -71 over 12. I hope friends you will like this video.

[00:13:32]Thank you so very much for watching.

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