Solving a 'Harvard' University entrance exam question | x=?

Added: 2026-06-02

[00:00:00]Hello everyone. You are welcome. Today we have a very interesting algebra math problem. Here is x squared divided by 25 is equal to 25 divided by x squared.

[00:00:12]So then we try to find out the values of x.

[00:00:15]So let's start our solution. So how can we solve this math problem? First of all, here we can cross multiply both the both sides, both expressions.

[00:00:24]So this will become This will become x squared times x squared is equal to 25 times 20 five.

[00:00:37]So here this x squared is two times, so therefore this will become this is x squared whole squared is equal to and this will become 25 whole squared.

[00:00:50]Here we can write this 25 as that is simply five squared whole squared.

[00:00:57]Here we will take this five squared whole squared to the left hand side. So this will become This is just x squared whole squared minus five squared whole squared is equal to zero.

[00:01:13]Here look at the left hand side here this is an algebraic identity a squared minus b squared.

[00:01:19]So here we will use this algebraic identity.

[00:01:23]a squared minus b squared that is equal to a plus b times a minus b.

[00:01:34]So using this algebraic identity here our a is simply x squared and our b is five squared.

[00:01:40]So this will become this is x squared plus five squared times x squared minus five squared is equal to zero.

[00:01:53]Here the product of these two quadratic algebraic identities is equal to zero.

[00:01:58]So, here either this expression will be zero or this one will be zero.

[00:02:03]So, from here we will get x squared plus 5 squared is equal to zero.

[00:02:10]Or x squared minus 5 squared is equal to zero.

[00:02:17]Now, first we will try to solve this one equation.

[00:02:19]So, here this left-hand side is in this one form a squared plus b squared that is equal to a plus bi times a minus bi.

[00:02:34]So, using this algebraic identity, here this expression will become here our a is x and our b is 5. So, this will become x plus 5i times x minus 5i is equal to zero.

[00:02:52]Here the product of these two linear expression is zero. So, here either this will be zero or this will be zero.

[00:02:58]So, from here we will get x plus 5i is equal to zero.

[00:03:03]Or x minus 5i is equal to zero.

[00:03:10]Here we will take this 5i to the right-hand side, so this will become x is equal to negative 5i.

[00:03:17]>> [snorts] >> And this will become here this is x is equal to this will become further to 5i.

[00:03:23]Here we get two values of x that are complex values.

[00:03:26]>> [snorts] >> Then we will try to solve this one equation x squared minus 5 squared.

[00:03:32]So, here our equation is simply x squared minus 5 squared is equal to zero.

[00:03:39]Now, this is algebraic identity a squared minus b squared, so this will become x minus 5 times x plus 5 is equal to zero.

[00:03:51]And from here we can write x minus five is equal to zero or x plus five is equal to zero.

[00:04:01]And we'll take this five to the right-hand side.

[00:04:04]So it become x is equal to five.

[00:04:07]And this will become x is equal to negative five.

[00:04:11]So from here we get two real solutions.

[00:04:14]So finally here we have four possible values of x which are x is equal to negative five positive five negative five i and positive five i.

[00:04:31]So finally here we have these four possible values of x.

[00:04:35]And we'll try to verify these values of x one by one. That is these values of x are the exact and correct value of x and verify this one equation or not.

[00:04:46]So let's try to verify these values here.

[00:04:48]Here first we'll try to verify x is equal to five.

[00:04:52]So here it is very simple for x is equal to five.

[00:04:56]If you want to verify x is equal to five, so here it is very easy. Here this will become five squared and this will become also five squared. Now five squared is 25 and five squared is 25.

[00:05:07]Both sides will be the same. 25 divided by 25. So here x is equal to five verify this one equation very easily.

[00:05:14]And we'll try to verify x is equal to negative five.

[00:05:18]So let's try this value. Here this will become here our x is minus five whole squared divided by 25 is equal to and the right-hand side will become 25 divided by our x is minus five whole squared.

[00:05:37]And we can write this negative five squared as negative five squared is simply this is five times negative one whole square.

[00:05:50]And we'll take this square over 5 and this -1. So, here 5 squared is simply this is 25.

[00:05:57]And here -1 squared it is simply -1 * -1 is positive 1. So, this become just 1.

[00:06:03]And 25 * 1 is 25. So, therefore the value of -5 squared is simply 25.

[00:06:10]So, this will become 25 / 25.

[00:06:15]Is equal to and this will be as become 25 / 25.

[00:06:20]And we can cancel 25 by 25 in both sides. So, this will become just 1 is equal to 1.

[00:06:28]Since both sides are equal, so it's mean that x is equal to -5 also verify this one equation.

[00:06:35]I will try to verify the complex values that are positive 2 5i and -5i.

[00:06:42]So, first we will try to verify positive 2 5i.

[00:06:46]So, the equation will become x is positive 5i whole square divided by 25 is equal to 25 divided by 5i whole square.

[00:07:03]So, here we will use this one algebraic identity exponential identity a * b whole square is equal to it is a squared * b squared. So, using this identity here in the numerator this will become this is just 5 squared * i squared divided by 25.

[00:07:23]And this will become 25 divided by this will become 5 squared * i squared.

[00:07:31]Here we know that i squared it is simply -1.

[00:07:35]So, therefore this equation will become 5 squared is 25. So, it become 25 * -1 / 25 = This will become 25 / 5 ^ 2 is 25 * -1.

[00:07:53]And we can cancel 25 and 25 and 25 and 25.

[00:07:57]This will become -1 / +1 and this will become 1 / -1.

[00:08:03]And we know that -1 / 1 is 1.

[00:08:08]Sorry, -1. = and 1 / -1 is also -1.

[00:08:14]And both sides are equal, so it means that x = 5i also verify this one equation.

[00:08:21]Now, finally we will try to verify the negative complex value, that is x = -5i. That is the last value.

[00:08:32]So, [snorts] let's substitute this value in the equation.

[00:08:34]The equation is This is -5i ^ 2 / 25 = 25 / -5i ^ 2.

[00:08:49]So, further we can write this as This will become We can write this as -5 ^ 2 * i ^ 2 / 25.

[00:09:01]And this will become 25 / -5 ^ 2 * i ^ 2.

[00:09:09]And -5 ^ 2 is simply -5 * -5, which is simply 25.

[00:09:15]* and i ^ 2 is simply -1 / 25.

[00:09:21]= This is 25 / This is 25 * -1.

[00:09:29]Again, here we can cancel this negative this 25 by 25 and this 25 by this 25.

[00:09:35]This will become -1 / 1 and this will become 1 / -1.

[00:09:40]So, again -1 / 1 is simply -1 = and 1 / -1 is again -1.

[00:09:47]Here both sides are equal, so it's mean that x = -5 I also verify the exponential algebraic equation.

[00:09:56]So, finally x = -5 +5 -5 i +5 i of the exact and correct [snorts] solutions of this interesting algebraic math problem.