This System Stumps 95% of Students! | Math Challenge

Added: 2026-05-12

[00:00:02]Hello friends, welcome back to NC John.

[00:00:04]Today in this video we'll be solving one very interesting system of equation problem for the real values of X and Y.

[00:00:12]So, let's get it started by writing here equation number one.

[00:00:17]And this equation we will call equation number two.

[00:00:22]Now, we are going to subtract second equation from first one.

[00:00:27]So, we will write here 1 - 2.

[00:00:31]So, we are going to get X - Y square root X square - Y square will get over will be equal to 5 - 3.

[00:00:43]Or we can write X - Y will be equal to 2.

[00:00:49]So, from this equation we will get Y in terms of X Y equal to X - 2.

[00:00:57]Let's say this is our equation number three.

[00:01:01]Now, we are going to use equation one and equation three.

[00:01:08]We will write X + the square root of X square - Y square equal to 5.

[00:01:18]This is our equation one.

[00:01:20]Now, we'll use equation three and write X + the square root of X square Y is X - 2. We'll write X - 2 whole square.

[00:01:36]Will be equal to 5.

[00:01:40]Now, we'll be using A - B whole square identity over here.

[00:01:45]So, let me write A - B whole square equal to A square 2 * AB + B square.

[00:01:56]Which we'll apply here and we will get x plus the square root of x square minus x minus two whole square is x square minus four x plus four.

[00:02:15]And RHS is five.

[00:02:19]We'll write x plus the square root of x square minus x square plus four x minus four equal to five.

[00:02:35]Now, we'll cancel x square with minus x square.

[00:02:39]So, this will give us x plus the square root of four x minus four equal to five.

[00:02:49]Now, we'll be subtracting x from both the sides.

[00:02:53]So, we'll write minus x in RHS.

[00:02:56]Plus and minus x will get over from left hand side. We will get the square root of four x minus four equal to five minus x.

[00:03:10]Now, we have a square root in LHS. So, we are going to a square both sides.

[00:03:16]But, before that, if we use basic algebra then we have two conditions from this equation if x has to be real. Condition number one radicand must be non-negative.

[00:03:32]So, we will write here four x minus four should be greater than or equal to zero.

[00:03:40]Condition number two, a square root must be non-negative.

[00:03:46]So, five minus x equal to a square root.

[00:03:49]So, this value must be greater than or equal to zero.

[00:03:53]So, we'll write our second condition, 5 - x should be greater than or equal to zero.

[00:04:02]So, from first inequality we will write 4x should be greater than or equal to 4.

[00:04:10]Or we will write x should be greater than or equal to 1.

[00:04:16]From left-hand side.

[00:04:18]And from right-hand side, 5 - x is greater than or equal to zero. So, from here we will get x should be less than or equal to 5.

[00:04:29]So, there are two conditions which x has to satisfy. So, we will get x should be in the interval 1 to 5.

[00:04:41]Our x value, if it is in in the interval 1 to 5 then we'll say our solutions are real.

[00:04:50]Now, we'll be squaring both sides.

[00:04:54]We will write square root of 4x - 4 equal to 5 - x.

[00:05:07]We'll put power two. We'll put power two.

[00:05:11]Now, a square root and a square will get over from LHS.

[00:05:16]And we'll be using a minus b whole square formula in RHS.

[00:05:23]So, we can write 4x - 4 equal to 5 square is 25 minus 2ab, two times five times x, minus 10x plus x square.

[00:05:41]Now, we'll take all the terms to RHS.

[00:05:44]So, we will get x is square -10x -4x -14x 25 + 4 29 equal to zero.

[00:05:59]Now, we have one quadratic equation. We can use quadratic formula.

[00:06:05]So, I need to write here a value which is coefficient of x is square one.

[00:06:10]B value which is coefficient of x -14 and c will be equal to constant 29.

[00:06:20]As per formula, x will be equal to -b plus minus the square root of b square minus four times a times c over two times a.

[00:06:36]Let's plug in all the values of a, b, and c. We will get minus of minus 14 +14 plus minus the square root of -14 is square.

[00:06:52]It is 196.

[00:06:54]minus four times one times 29 will be equal to 116 over two times one will be two.

[00:07:07]So, we can write x will be equal to 14 plus minus the square root of 196 minus 116 80 over two.

[00:07:22]Now, we can factor 80 as 16 times five.

[00:07:27]So, x will be equal to 14 plus minus the square root of 16 times five over two.

[00:07:40]Now, the square root of 16 four.

[00:07:44]So, we will write 14 plus minus four times the square root of five over two.

[00:07:53]Which will give us seven plus minus two times the square root five.

[00:08:01]So, we have two x values.

[00:08:04]Seven plus two square root five.

[00:08:09]And second is seven minus two square root five.

[00:08:14]Square root five is approximately 2.23.

[00:08:19]So, two times 2.23 is 4.46.

[00:08:23]So, 7 plus 4.46 approximately 11.46.

[00:08:31]And 7 minus 4.46 will be approximately 2. 5 4.

[00:08:40]Now, our condition on x is x should be in the interval one to five.

[00:08:47]So, we are going to reject seven plus two square root five as it is greater than five.

[00:08:55]So, we can accept 2.54 which is seven minus two square root five.

[00:09:03]So, x will be equal to seven minus two square root five.

[00:09:10]Now, we have to find y value.

[00:09:13]And y is x minus two.

[00:09:18]We'll put the value of x seven minus two square root five minus two.

[00:09:26]So, 7 minus 2 is five. We are going to get y value five minus two square root five.

[00:09:37]So, x is seven - 2 square root 5.

[00:09:40]And Y is 5 - 2 square root 5. I hope friends you will like this video. Thank you so very much for watching. Do not forget to like, share, and subscribe.

[00:09:51]Bye-bye till next video. Good luck.