Germany | Can you solve this ? | A Nice Maths Olympiad Algebra Problem

Added: 2026-05-10

[00:00:00]Hello, you're welcome to solve this math problem which is m * m * m - m * m is = 100. to find the values of m from this equation.

[00:00:15]Now solution in the first step from here m * m * m is same as m^ of 3. Then minus m * m is same as m^ 2 is equal to 100.

[00:00:32]Then from here take 100 to the left side. So it will be m^ 3 - m^ 2. When we take this side will be minus 100 is = 0.

[00:00:46]Then here it will be m^ of 3 - m^ 2 minus 100 we split into a into numbers which are in difference which are in difference in form of exponents. So 100 is same as 125 - 25 bracket is equal to this zero.

[00:01:15]Then here it will be m^ 3 - m^ 2 we take negative inside. So it will be -25 minus a minus it will be + 25 is equal to zero.

[00:01:32]Then here it will be m^ 3 - m^ 2 - 125 into exponents it is 5^ 3 + 25 into exponents it is 5^ of 2 is equal to zero.

[00:01:52]Then with like powers of three we place together. So we start by this m^ 3 then - 5^ 3 then here with like powers of two together. So this here which is - m^ 2 then + 5^ 2 is equal to zero.

[00:02:16]Then we place inside the bracket. So here it will be m^ 3 - 5^ 3 inside the bracket into here we take negative outside the bracket. Now m² / it is positive m² / is - 5² bracket is equal to this zero.

[00:02:46]Then from this part here this part m^ 3 - 5^ 3 is in the form of the rule which is a power 3 - b^ 3 which is equal to a - bracket bracket a² + a b + b square bracket and from this part here m² - 5 square is in the form of the rule which A² - B² is equal to A - B bracket bracket A + B bracket.

[00:03:29]Then from here from this part we apply this form. So a minus b it will be m - 5. So here it will be m - 5 bracket bracket a² it will be m² then plus a b it will be m * 5 is 5 m + b² it will be 5² it is 25 bracket then minus into this part is in this form then we change into this form so it will be a minus b is m - 5 so Here bracket m - 5 bracket bracket a + b it will be m + 5.

[00:04:13]So here m + 5 bracket is equal to this zero.

[00:04:21]Then in the next step from this equation m - 5 it is common. So we take m -5 bracket outside the [snorts] bracket.

[00:04:33]Now this here / mus 5 it is this. So it is this here which is m² + 5 m + 25.

[00:04:47]Then here we have this negative. Then this divide by this it is this here with m + 5. So here bracket m + 5 bracket bracket is equal to zero.

[00:05:04]Then into here it will be m - 5 bracket then bracket here it is m² + 5 m + 25 we take negative inside to be - m here it will be - 5 bracket is = 0 then here to be m - 5 bracket bracket this Here m² 5 m - m it is 4 m. So here + 4 m 25 - 5 is 20. So + 20 bracket is = 0.

[00:05:50]Then into here we have two solutions where this first solution of m - 5 is = 0. And we have this solution here. m² + 4 m + 20 is = 0. Then from this solution we take m to the5 to this side. So it will be m is = 5. So this is the first solution which is real solution. Now to solve from this quadratic equation we solve by using quadratic formula. Whereas coefficients A is equal to cofficient of M² is 1. B is equal to cofficient of M it is 4 and C is equal to constant it is 20. Now by applying quadratic formula to find the values of M is equal to B plus or minus square root of B² - 4 A C over 2 A. So into here it will be m is = b is 4 + or minus square roo<unk> of b² it will be 4² - 4 * a it is 1 * c it is 20.

[00:07:12]Then over 2 * a it is 1.

[00:07:18]Then in the next step it will be m is = -4 + or minus square<unk> of 4² it is 16. Then minus 4 * 20 it is 80. Then over 2 * 1 it is 2.

[00:07:38]Then into here it will be m is = -4 + or minus square<unk> of 16 - 80 it is - here 10 - 4 it is 10 - 6 it is 4 to be 7 - 1 it is 6 then over 2 then here it will be m is = -4 + or minus as square<unk> of -64 is same as 64 * -1 then over two then into here it will be m is = -4 + or minus here we separate the square roots so it will be square root of 64 *<unk> of -1 then over 2 then into here it will be m is = to -4 + or minus<unk> 64 it is 8<unk> of -1 it is iota then over this two then here it will be m is equal to we divide by 2 in here and here so it will be this over this so -4 / 2 + or minus 8 I / 2 so into here will be m is = -4 / 2 it is -2 + or minus 8 I / 2 it is 4 I. So from here we have two complex solutions.

[00:09:20]Now our conclusion the first value of M is equal to this real solution here which is five.

[00:09:32]The second value of x is equal to from complex solutions when it is positive to be -2 + 4 i. So here -2 + 4 i. The second the third value of x is equal to when it is negative it will be -2 - 4 i.

[00:09:54]So these are all the values of m from this our problem. We have three values of n. One real solution and two complex solutions.

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[00:10:16]Bye-bye.