Only 5% Can Solve This Challenging Algebraic Equation!

[00:00:02]Hello everyone, welcome to Infyan.

[00:00:05]Today we are going to solve one very interesting and challenging equation from algebra. Here we have to find all the real x. So let's get started by checking our equation at x=0.

[00:00:21]I will find left hand side at x=0.

[00:00:24]So 0² + 1 is 1 * 0^ 4 - 4 * 0² that is 0 + 1 is 1. So LHS will become 1 * 1 1.

[00:00:37]Now we will find RHS at x =0 6 * 0 cube 6 * 0 RS will be 0 1 cannot be equal to 0. So we will write here x cannot be zero for the given equation.

[00:00:56]Now we will divide both sides by x cq so that we can cancel x cube from the right hand side and in left hand side we will divide our bracket x² + 1 by x and second bracket x^ 4 - 4x² + 1 by x² x² x is x + 1 /x X * X^ 4x² is x² - 4 x² / x² is -4 + 1 / x² = x cq x cub will get over we will write 6 only.

[00:01:46]This will be written as x + 1 /x * x² + 1 / x² - 4 can be written as 2 - 6 = 6. Now we know that x² + 1 / x² + 2 is x + 1 /x².

[00:02:20]So our equation will become x + 1 /x* x + 1 /x² - 6 = 6.

[00:02:42]Now our substitution is obvious. Let us assume x + 1 /x suppose u.

[00:02:53]So we will write our equation u * u² - 6 = 6.

[00:03:04]By writing all the terms to one side you will get u cub - 6 u - 6 = 0.

[00:03:15]Now we are going to use another substitution.

[00:03:18]Let us assume a + b equal to u or u = a + b. So we will write our equation a + b whole cube minus 6 * a + b - 6 = 0.

[00:03:45]Now we are going to use one algebraic identity.

[00:03:50]a + b whole cube.

[00:03:55]This will be - 3 a b time a + b minus aq + bq.

[00:04:07]If we will expand a + b whole cube then we will get rh.

[00:04:14]Now we are going to compare coefficients from both the equation. I am presuming it is a + b variable equation. So we can write 6 = 3 a b and we can write 6 = aq + bq.

[00:04:35]Let me write here 3 a = 6 and second equation is aq + bq = 6.

[00:04:49]Now we will divide first equation by 3 both sides.

[00:04:55]6 / 3 is 2. 3 / 3 is 1. We will get a = 2.

[00:05:05]And here we have aqu + b cq = 6.

[00:05:12]I will be cubing first equation a = 2.

[00:05:16]So I will put three over here. Part three over here. Now we will have aqu * bq equal to 2 cq is 8.

[00:05:30]Second equation is aqu + b cq. This is equal to 6. So here we have aqu + bq.

[00:05:41]And here we have aqu * bq.

[00:05:45]If we will think about one quadratic equation suppose in variable t whose roots are a cube and b cq then we can write t² - 60 + 8 =0.

[00:06:04]After factorization we can easily write t - 2 * t - 4 =0.

[00:06:14]Using product zero rule we will get either t - 2 =0 or t - 4 = 0. So we'll have t = 2 t = 4. Now t is a cube or bq.

[00:06:35]So I am considering 2 = a. So I will write a = 2, b cq = 4.

[00:06:46]For real values of a and b, we will take cube root. A will become cube root of 2.

[00:06:55]B will become cube root of 4.

[00:07:01]Now we can easily find u which is sum of both a + b. a is cube <unk>2 b is cube roo<unk>4.

[00:07:15]U was our substitution. x + 1 /x was u.

[00:07:20]So I will write x + 1 /x = u. x cannot be zero. So we can easily multiply both sides by x.

[00:07:30]We will get one quadratic x² + 1 = ux by writing all the terms to one side. x² - ux + 1 = 0.

[00:07:48]Now we will use quadratic formula straight away. So we will get x in terms of u. So x will become minus b. So minus of - u + u + - square root of - u² is u² - 4 * 1 * 4 it is 4 only over 2 * 1 it is 2.

[00:08:15]Now we have to plug in u = cube <unk>2 + cube roo<unk>4. If we will apply both the values I will get u value approximately 2.8473 which we are going to apply here to get the real values of x.

[00:08:38]So let me write x = u + - square<unk> of u² -4 / 2 and u is 2 84 73.

[00:08:59]Let's put all the values. We will get x = 2.8473. 8473 + minus square t of 2.84 73 a² - 4 / 2 will be equal to 2.8473 + minus square root of 8.104 104 - 4 I am writing here square<unk> 4.104 over 2 which will be equal to 2.8473 + 2.026 / 2 with plus sign we are going to get 2.436 436 with minus sign in between we will get 0.410.

[00:10:08]So we have two real x which will satisfy our equation. I hope friends you will like this video. Thank you so very much for watching. Do not forget to like, share and subscribe. Bye-bye till next video. Good luck.