Olympiad Mathematics | Indian | Can You Solve This? | The Four Solutions

本站添加: 2026-05-14

[00:00:01]If you are ready, let's solve this one very quickly.

[00:00:06]But mind you, we are going after the complete solution.

[00:00:11]Solution we have x ^ 2 over 25 to be equal to um 25 over x ^ 2.

[00:00:31]Okay. And I'm thinking how do I solve this?

[00:00:36]Since we are to solve it completely, we cross multiply. That will be the first step.

[00:00:43]We have x^2 * x 2 to be = 25 * 25.

[00:00:51]And um we know that x 2 * x^2 is the same as x ^ 4. And then here 25 is 5 2 and then another 25 again which is still 5 squared.

[00:01:09]And from one of the laws of indices x ^ 4 will be pick one of the bases add the powers and we get four.

[00:01:21]Now we're not stopping here because we want to get the four solutions. So we will now do x ^ 4 - 5 ^ 4 and is equal to zero.

[00:01:37]x ^ 4 - 5 ^ 4 to be = 0. Now what do we do? We have difference of um two numbers raised to the^ 4. But here is what we are going to do. We can reduce it to difference of two squares.

[00:01:58]That means that we write x² 2 - 5 2 then we square this again and all is equal to zero. So by now you can see your difference of two squares clearly.

[00:02:14]So we are writing x² - 5 2 is 25. So we have 25 squar now to be equal to zero.

[00:02:25]And applying our popular difference of two squares we will now have remember that a^ 2 minus b² is a - b into a + b.

[00:02:39]I believe we know that. So that means that the difference of two squares are both now will be um x^2 - 25 into x^2 + 25 and everything is equal to zero.

[00:02:58]So from here now we have to settle down and solve it because we are going to apply another rule which we call zero product rule.

[00:03:12]We apply this rule when we multiply two terms like this to get zero. So either of them must be equal to zero. [snorts] Just like saying you have um a * b to be zero. Okay, it means that it is either a is 0 or b is 0. If a is 0, 0 * b is 0.

[00:03:37]And if it is b that is zero, a * 0 will still give z, right? So at this point, we are very sure that we applying the same thing to this. So to do that we will have our x² - 25 to be = 0 or x² + 25 to be = 0.

[00:04:05]Okay. So let's pick it up from here.

[00:04:09]Okay. So from here now we we're going to solve this one first. So we have x² 25 is 5^ 2 and this is equal to 0. I'll get back to that. Mind you we now have difference of two squares here and the difference can okay we've already done something like this before when I said a^ 2 - b^ 2 is equal to a minus b * a + b. So the same thing is happening here.

[00:04:42]So that we can have a okay x - 5 then multiply by x + 5 and this is now equal to zero. So because we are multiplying two numbers or two terms to get zero we have to apply zero product rule again saying that x - 5 is 0 or x + 5 is zero.

[00:05:11]Now from the from here our x is going to be 0 + 5 or from here x is 0 - 5. So our x now is five from here x is 5 or x is what - 5. As a matter of fact we have two solutions from here.

[00:05:37]X1 is 5, X2 is -5.

[00:05:42]But then we left out one of the let's look at this one of the factors and that is um x^2 + 25 = 0. So I'm going to bring that down right now.

[00:05:56]We have x² + 25 to be equal to zero. So what do we do in this case? We're going to take 25 to the other side of the equation and it gives us x² to be equal to - 255.

[00:06:17]So this means that um we're going to get complex solution from here because of the - 255.

[00:06:26]Take the square root of both sides of the equation.

[00:06:30]Square root will help us to eliminate the square. Now on the right we have plus or minus the square root of 20.

[00:06:38]It's the square root of -25.

[00:06:43]So this one and this one are going to go.

[00:06:47]Now we have x alone to be equal to plus or minus roo<unk> of 25. Oh there's negative right? So that means we multiply this by1 so that we can easily find the square root of 25. So our x is plus or minus square root of 25 is 5 and square root of -1 is imaginary. So we write I multiply the two so that x will be plus or minus square roo<unk> of 5 um 5 * i rather and that will be 5 i and this is a two in one solution already because from here again we have our x okay we are having our x to be equal to 5 i or -5 I. So two more solutions from here.

[00:07:45]We're going to bring the four solutions together right away.

[00:07:50]Okay. So from the first part we have that x is = 5 and we equally have that x is = -5. So this is our x1 our x2.

[00:08:05]Now from the second part from the second part we have that x 3 will be 5 I and then the fourth solution x4 will be equal to -5 I. So these are the four solutions to the equation.

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