Only 1% Solve this German Math Olympiad | Can You Solve This One?

[00:00:00]You're welcome to solve this nice algebra problem which is 2x + 13^ 2 / x + 7 ^ 3. This is equal to 640. So what is the value of x given that x is an element of real numbers. So let's provide a solution from here.

[00:00:23]Now we have 2x + 13 raised to ^ of 2. This is divided by x + 7 raised ^ of 3. This is equal to 640.

[00:00:40]Now the first step to do here let's multiply on both sides by 1 / 2 ^ of 3.

[00:00:46]Here we have 1 / 2 ^ of 3. So that now here we have 2x + 13 raised to ^ of 2 then dividing by 2 ^ of 3 * x + 7 ^ of 3 this is = 640 dividing by 2 ^ 3 2 ^ 3 in this case this is 8.

[00:01:18]So that now we have 2x + 13 raised to ^ of 2 this is divided by now from what we have here this is 2x + 2 * 7 this is 14 raised to ^ of 3 this is equal to 64 / 8 this is equal to 80 so we have 80.

[00:01:48]In the next step, we have in the numerator 2x + 13 raised ^ 2. This is divided by 2x + 14.

[00:02:00]This is the same thing as 2x + 13 + 1.

[00:02:05]This is raised to ^ of 2 of 3. And this is equal to 80.

[00:02:13]So we have 2x + 13 is common here. We can let 2x + 13 be = k. So let's substitute k here so that we have k² in the numerator divided by k + 1 raised to ^ 3.

[00:02:36]This is equal to 80. This is equal to 80. So now we have k² / k + 1 ^ 3. In the next step we can write k b = 1 / u. So let's substitute 1 / u here in the numerator. This is 1 / u 2 [snorts] / 1 / u + 1. This is raised to the power of 3. And this is equal to 80.

[00:03:14]So we have 1 / u + 1. Remember 1 is our whole number. So this is over 1. LCM is u. Uid u is 1 * 1. We have 1 + u / 1 is u * 1. This is u.

[00:03:33]And therefore we have 1 / u^ 2 / u + 1 / u. This is raised to the power of 3.

[00:03:47]And therefore this is equal to 80.

[00:03:52]Now we have 1 / u^ 2 / we can express this as u + 1 ^ 3 u + 1 ^ 3 / u ^ 3 this is equal to 80.

[00:04:11]In the next step we have 1 / u^ 2 * u ^ 3 / u + 1 raised ^ 3 this is equal to 80.

[00:04:26]Let's simplify here. U²id u this cancels out u ^ 3 / u square this is u. So we have u / u + 1 raised ^ 3. This is equal to 80. This is equal to 80. In the next step in the numerator here we have u.

[00:04:51]Let's introduce + 1 - 1 / [snorts] u + 1 raised to ^ 3. This is equal to 80.

[00:05:03]Now we can split this as u + 1 / u + 1.

[00:05:11]This is raised ^ of 3. Then subtract 1 / u + 1 raised ^ 3. And this is equal to 80.

[00:05:24]Now we can simplify here. This is u + 1 raised to the power of 1. Remember if you have a ^ n / a the^ m this can be expressed as a ^ n minus m.

[00:05:41]Applying this exponent property then we have 1 / u + 1.

[00:05:49]This is free to the power of 2. Then - 1 / u + 1 ^ of 3. This is equal to 80.

[00:06:02]Now we have that this is the same thing as 1 ^ 2. If you say 1 2 this is the same thing as one. If we have 1 ^ 3 this is the same thing as 1. So we have 1 / u + 1 raised ^ of 2 subtract 1 / u + 1 raised ^ 3. This is equal to 80.

[00:06:30]We have 1 / u + 1 is common here. So we can let 1 / u + 1 be = n. Let's express this equation in terms of n. So that we have n² - n ^ 3. This is equal to 80. Let's take 80 on the left hand side. So that we have n^ 2 - n ^ 3 - 80 this is = 0. So let's in the next step let's rearrange this equation here so that we have - n ^ 3 + n^² - 80 this is equal to 0. The next step is to multiply both sides by -1. And therefore we have n ^ 3 * -1.

[00:07:24]This is n ^ 3 subtract n² then + 80. This is equal to 0.

[00:07:33]In the next step we have n ^ 3 - n ^ 2 + 80 which is 64 + 16 and this is equal to zero.

[00:07:46]So we have n ^ 3 - n ^ 2 + 64 this is 4 ^ 3 then + 16 which is 4 ^ 2 and this is equal to zero.

[00:08:01]Now we have n ^ 3 and 4 ^ 3 here. So this is n ^ 3 + 4 ^ 3 - n ^ 2 + 4 ^ 2 this is equal to to zero. So this is n ^ 3 + 4 ^ 3 subtract into the parenthesis.

[00:08:24]This is n^ 2 - 4^ 2 and this is equal to 0.

[00:08:30]We have got two parts here. The first part is the sum of two cubes expressed as a ^ 3 + b ^ 3. The second part here, this is the difference of two perfect squares expressed as a square minus b².

[00:08:45]So a ^ 3 + b ^ 3. This can be expressed as a + b multiplying by a 2 - a b + b ^ 2.

[00:09:03]Applying this identity then here we have n + 4 multiplying by n² subtract 4 n then + 4 2 which is 16.

[00:09:19]Then we have subtract the difference of two perfect squares here. This is the same thing as a + b multip by a subtract b.

[00:09:31]So in other words we have here - n + 4 then we have n - 4 this is equal to 0.

[00:09:41]So we have n + 4 is common here. Let's factor out n + 4 into the parentheses.

[00:09:49]This is n² - 4 n then + 16 then subtract n - * -4 this is + 4 and this is equal to zero. So we have n + 4 * n^ 2 - 4 n - n this is - 5 n 16 + 4 this is + 20 and this is equal to zero.

[00:10:24]Now we have two possible cases here. The first case is n + 4. This is equal to z.

[00:10:31]The second case is n² subract 5 n + 20 this is equal to zero.

[00:10:41]So we have n is = -4.

[00:10:47]And on the second part here, this is a quadratic equation where we have a is = 1, b is = 5 and c is = to 20.

[00:11:03]Now in the next step, let's assess the nature of the root of this quadratic equation here by determining the discriminant value which is b ^ 2 - 4 a c. Now let's substitute a, b and c. We have -5 raised ^ 2 subtract 4 * 1 * c which is 20. And therefore we have this is 25 minus 4 * 20. This is 100. And therefore we have 25 - 1. This is 4 * 20. This is 80. Sorry 25 - 80. This is equal to 55. And this is less than zero. So in other words, we have two complex root.

[00:11:51]We have two complex roots here. And this indicates that there is no real solution. There's no real solution that can be obtained from this quadratic equation. So in other words, this quadratic equation here is rejected.

[00:12:12]Now let's focus on n which is = -4.

[00:12:17]We have n is =4.

[00:12:20][snorts] Now if you recall if you recall we stated that we let 1 / u + 1 be = n.

[00:12:37]So this is to mean that we have 1 / u + 1 this is equal to -4 -4 is our whole number. So this is over 1. If we cross multiply here we have -4 * u + 1. This is = 1. So this is -4 u - 4. This is equal to 1. Let's take4 on the right hand side. So we have -4 U. This is = 1 + 4 and this is 5.

[00:13:09]Let's divide on both sides by -4.

[00:13:12]And we have that U is = -5 / 4. U is = -5 / 4.

[00:13:24]Now since we have u is = - 5 / 4. And if you recall we stated that we let k be = 1 / u. So this means that k is = 1 / -5 / 4. And this means that k is the reciproc which is -4 / 5. This is the value of k.

[00:13:54]Again if you recall we in that we let 2x + 13 be = k.

[00:14:05]Let's substitute k so that we have 2x + 13. This is = -4 / 5. So 2x + 13 this is the whole number. So this is over 1. Now let's cross multiply from here. So that we have 5 * 2x + 13 this is equal to -4.

[00:14:30]So this becomes 5 * 2x this is 10 x + 65 this is = -4.

[00:14:38]Let's take + 65 from the right hand side. So that we have 10 x this is = -4 - 65 and this means we have 10 x this is = 69.

[00:14:58]Let's divide on both sides by 10.

[00:15:01]And this means that the value of x is equal to6.9.

[00:15:10]And this is the solution to this algebra problem. Now let's check if this value of x satisfies the equation.

[00:15:20]Now let's check if this value of x which is - 6.9. If this satisfies the equation.

[00:15:27]Now if you recall from the equation we have 2 that is 2x + 13 raised to the power of 2. Then dividing by x + 7 raised to the power of 3.

[00:15:46]This should give us a value of 6 40.

[00:15:52]Substituting x here this means we have 2 [snorts] * -6.9 + 13 this is raised to ^ 2 then divided by -6.9 + 7 raised to ^ 3. This should give us a value of 640.

[00:16:18]In the next step, we have 2 * - 6.9. This is the same thing as 13.8 + 13 raised to ^ 2. Then divided by -6.9 + 7. This is the same thing as - 0.1 raised to the^ of 3. And this should give us a value of 640.

[00:16:52]So -3.8 + 13 this is equal to 0.8 raised ^ 2. Then this is divided by we have 0.1 raised ^ 3. This is the same thing as 0.001.

[00:17:13]So this should be equal to 640.

[00:17:17]In the next step we have0.8 ^ 2. This is the same thing as 0.64 divided by 0.001.

[00:17:30]This should be equal to 640.

[00:17:33]Now 0.64 over this this is the same thing as 640 which is equal to 640.

[00:17:42]So the left hand side is equal to the right add side and this affirms that the value of x here which is -6.9 satisfies the equation. So kindly like this video, share and subscribe. See you in the next video.