Germany | Can you solve this? | Math Olympiad

Added: 2026-05-09

[00:00:00]Hello everyone. Welcome to how to solve this very nice quartic equation. X to the power 4 plus 8 X minus 7 equal to 0.

[00:00:09]Our job is to find all possible values of X. So, let's start.

[00:00:14]If we divide this constant 7 by the leading coefficient 1, if we divide 7 by 1, we get 7. And factors of 7 are plus minus 1 and plus minus 7.

[00:00:30]If we check a plus minus 1 in this equation, plus minus 1 is not a solution. If we check a plus minus 7 in this equation, plus minus 7 is not a solution.

[00:00:43]So, let's use another trick to solve this equation.

[00:00:47]To make a perfect square at left-hand side, we add and subtract 2x² and we add and subtract 1.

[00:00:58]So, this will become x to the power 4.

[00:01:03]We add 2x² and we subtract 2x².

[00:01:11]And we add 1 and we subtract 1.

[00:01:16]These remaining two terms, 8x - 7, plus 8 * x 7 equal to 0.

[00:01:26]Now, this x to the power 4 can be written as x² whole squared. Plus this 2x² can be written as 2 * x² times 1.

[00:01:40]Plus this 1 can be written as 1² and we combine these terms in -2x² and + 8x, -2x² + 8 * x.

[00:01:56]This -1 and -7 will become negative eight equal to zero. By using this algebraic identity a squared plus two times a times b plus b squared equal to a plus b whole squared. This x squared squared plus two times x squared times one plus one squared will become x squared plus one whole squared. And from these three terms a negative two x squared plus eight x and negative eight, we can factor out negative two. In bracket left x squared minus four times x plus four equal to zero.

[00:02:51]Next, x squared plus one whole squared minus two times This x squared minus four x plus four is a perfect square. This can be written as x squared minus two times x times two plus this four can be written as two squared equal to zero.

[00:03:23]By using this algebraic identity a squared minus two times a times b plus b squared equal to a minus b whole squared.

[00:03:37]This expression x squared minus two times x times two plus two squared will become x minus two whole squared.

[00:03:50]And this a negative two times a negative two and this expression x ^ 2 + 1 whole ^ = 0.

[00:04:03]Now, we can rewrite this expression as x ^ 2 + 1 whole ^ root 2 * x - 2 whole ^ = 0.

[00:04:26]Now, we have difference of two squares, so by using this algebraic identity a ^ 2 - b ^ 2 = a + b * a - b.

[00:04:41]This expression will become x ^ 2 + 1 + root 2 * x 2 * x ^ 2 + 1 root 2 * x - 2 = 0.

[00:05:09]Further simplified, this will become x ^ 2 + 1 root 2 * x root 2 x and root 2 * -2 -2 * root 2 * x ^ 2 + 1 - root 2 * x - root 2 x and - root 2 * -2 + 2 * root 2 = 0.

[00:05:45]Now, we rearrange these terms, x squared plus root two times x plus one minus two times root two.

[00:05:58]Times rearrange these terms, x squared minus root two times x plus one plus two times root two equal to zero. Either this expression x squared plus root two times x plus one minus two times root two equal to zero or this expression x squared minus root two times x plus one plus two times root two equal to zero. Means that we have two quadratic equations and two cases.

[00:06:39]This is case one and this is case two.

[00:06:43]First, we solve this case one, case number one. In case one, we have this equation x squared plus root two times x plus one minus two times root two equal to zero.

[00:07:03]This is not factorable, so we solve it by quadratic formula. According to quadratic formula, x is equal to negative b is root two plus or minus the square root of in place of b squared, we write root two squared minus four times a is one times c is one minus two times root two divided by two times a is one.

[00:07:34]Now, this square will be canceled out with the square root and we are left with x is equal to negative root 2 plus or minus the square root of 2 negative 4 * 1 * 1 negative 4 and negative 4 * negative 2 * root 2 plus 8 * root 2 divided by 2 * 1 2 Further simplify, this will become x is equal to negative root 2 plus or minus the square root of We write this 8 * root 2 first 8 * root 2 and 2 - 4 will become negative 2 divided by 2. So, these are the two solutions from this case.

[00:08:32]Now, we move on to case number two.

[00:08:35]In case number two, we have this equation x squared minus root 2 * x + 1 + 2 * root 2 equal to 0.

[00:08:49]In case number two, we have the equation x squared minus root 2 * x plus 1 plus 2 * root 2 equal to 0.

[00:09:06]This is not factorable, so we solve it by quadratic formula.

[00:09:10]According to quadratic formula, x is equal to negative b is negative root 2 plus or minus the square root of In place of b squared, we write in negative root 2 squared minus 4 * a is 1 * c is 1 plus 2 * root 2 divided by two times a is one.

[00:09:42]Further simplify, this will become x is equal to positive root 2 plus or minus the square root of This is negative root 2 squared will become positive 2 and negative 4 times 1 times 1 negative 4 negative 4 times 2 times root 2 minus 8 times root 2 divided by 2.

[00:10:13]Further simplify, this will become x is equal to root 2 plus or minus square root of This is negative 8 times root 2 negative 8 times root 2 and 2 minus 4 will become negative 2 divided by 2.

[00:10:34]Next, x is equal to root 2 plus or minus square root of From these two terms, we can factor out a negative 1 and in bracket left 8 times root 2 plus 2 divided by 2.

[00:10:58]Since root negative 1 equal to i so this will become x is equal to root 2 plus or minus i times the square root of 8 times root 2 plus 2 divided by 2.

[00:11:21]So, from the second case, we get two complex solutions third and fourth value of x.

[00:11:29]And first we found that x 1 and two is equal to First we found that X is equal to negative root two plus minus the square root of eight times root two minus two all this divided by two.

[00:11:53]First we found that X is equal to negative root two plus or minus the square root of eight times root two minus two divided by two.

[00:12:10]So we have four solutions for this equation.

[00:12:14]These two solutions are real and these two solutions are complex.