It's funny how a simple-looking equation like x² - 4x + 1 = 0 can open up a whole world of mathematical exploration. At first glance, it's just another quadratic equation, something we might have wrestled with in high school algebra. The standard way to tackle it, of course, is using the quadratic formula or completing the square. And indeed, solving it directly gives us x = 2 + √3 or x = 2 - √3. These are the roots, the specific values of 'x' that make the equation true.
But what's truly fascinating is how this foundational equation becomes a springboard for so many other interesting problems. It's like finding a key that unlocks several doors, each leading to a different puzzle.
For instance, once we know x² - 4x + 1 = 0, we can start asking questions about expressions involving 'x'. Take the expression (x²)(x⁴ - 4x² + 1). It looks a bit intimidating, doesn't it? Yet, by cleverly manipulating the original equation, we can find its value. A common trick is to divide the original equation by 'x' (assuming x isn't zero, which it isn't for these roots). This gives us x - 4 + 1/x = 0, or more usefully, x + 1/x = 4. Squaring both sides of this new equation, (x + 1/x)² = 4², leads us to x² + 2 + 1/x² = 16, which simplifies to x² + 1/x² = 14. Now, if we look at the expression we need to evaluate, (x²)(x⁴ - 4x² + 1), and divide the second factor by x², we get x² - 4 + 1/x². We already know x² + 1/x² = 14, so x² - 4 + 1/x² becomes 14 - 4 = 10. Multiply that by the x² we factored out earlier, and we get 10x². Wait, that's not quite right. Let's re-examine the expression (x²)(x⁴ - 4x² + 1). If we distribute the x², we get x⁶ - 4x⁴ + x². This doesn't seem to directly use our x + 1/x = 4 relationship. Ah, but the reference material shows a different approach for a similar expression: (x⁴ - 4x² + 1) / x². If we divide the numerator by x², we get x² - 4 + 1/x². And we've already established that x² + 1/x² = 14. So, x² - 4 + 1/x² = 14 - 4 = 10. This means that if the question was asking for (x⁴ - 4x² + 1) / x², the answer would be 10. The reference material confirms this, showing that if x² - 4x + 1 = 0, then (x⁴ - 4x² + 1) / x² equals 10.
Another common twist is evaluating expressions like (x + 1/x)². We already found x + 1/x = 4, so (x + 1/x)² is simply 4² = 16. And if we're asked for x² + 1/x², that's precisely what we found: 14.
It's also interesting to see how the equation can be used to find the value of seemingly unrelated fractions. For example, if we're asked to find the value of (x + 2) / (4x + 2) given x² - 4x + 1 = 0, it's not immediately obvious. However, by dividing the numerator and denominator by 'x', we get (1 + 2/x) / (4 + 2/x). This still doesn't look easy. The reference material suggests a different path: dividing the original equation by 'x' gives x + 1/x = 4. Then, it seems there's a leap to calculating (x+2)/(4x+2). The provided solution states the answer is 1/10. The explanation mentions dividing the equation by 'x' to get x + 1/x = 4, then squaring to get x² + 1/x² = 14. It then says 'x² + = 14' and ' = 1/10'. This suggests a more complex manipulation or a different expression was intended in the explanation. However, the core idea is that the initial equation x² - 4x + 1 = 0 provides a fundamental relationship that can be exploited.
What about factoring? While x² - 4x + 1 doesn't factor nicely into simple integers, the related equation x² + 4x - 1 = 0 can be factored using the quadratic formula's roots. If we solve x² + 4x - 1 = 0, we get x = -2 ± √5. This means the factored form would be (x - (-2 + √5))(x - (-2 - √5)), which simplifies to (x + 2 - √5)(x + 2 + √5).
It's a testament to the elegance of algebra that a single quadratic equation can lead to such a variety of problems and solutions. It reminds us that understanding the fundamental relationships within an equation is key to unlocking more complex mathematical landscapes.