Solving X^4-10x^2+21=0 By Substitution A Step By Step Guide
In the fascinating world of algebra, we often encounter equations that seem daunting at first glance. However, with a little bit of cleverness and the right techniques, these equations can be tamed and solved. One such technique is substitution, a powerful method that allows us to simplify complex equations by introducing new variables. In this article, we'll delve into the method of substitution, using the equation $x^4 - 10x^2 + 21 = 0$ as our primary example. We'll explore the steps involved in solving this equation, highlighting the underlying principles and providing valuable insights along the way. So, let's embark on this mathematical journey and uncover the beauty of substitution!
Understanding the Equation: A Closer Look
Before we dive into the solution, let's take a moment to understand the equation we're dealing with: $x^4 - 10x^2 + 21 = 0$. This equation is a quartic equation, meaning it involves a variable raised to the power of four. Quartic equations can be tricky to solve directly, but notice something special about this one. The exponents of $x$ are even numbers (4 and 2), which suggests that we might be able to transform it into a more familiar form. This is where the method of substitution comes into play. By introducing a new variable, we can rewrite the equation in a way that makes it easier to solve. The key is to choose the right substitution, one that simplifies the equation and allows us to apply our existing knowledge of algebraic techniques.
The Power of Substitution: Simplifying the Complex
The beauty of substitution lies in its ability to transform complex equations into simpler ones. The core idea is to replace a complicated expression with a single variable, making the equation more manageable. In our case, we have $x^4$ and $x^2$ terms. A natural choice for substitution would be to let $u = x^2$. This substitution allows us to rewrite the equation in terms of $u$, effectively reducing the degree of the equation. So, if $u = x^2$, then $u^2 = (x2)2 = x^4$. Now we can rewrite our original equation using $u$ instead of $x^2$ and $x^4$. This transformation is the heart of the substitution method, and it's the first step in unraveling the solution to our equation. By choosing the right substitution, we pave the way for a more straightforward solution process.
Making the Substitution: Transforming the Equation
Now that we've identified the appropriate substitution, let's put it into action. We'll replace every instance of $x^2$ with $u$ and every instance of $x^4$ with $u^2$. Our original equation, $x^4 - 10x^2 + 21 = 0$, becomes: $u^2 - 10u + 21 = 0$. Notice how the substitution has transformed our quartic equation into a quadratic equation in terms of $u$. This is a significant simplification because we have well-established methods for solving quadratic equations. We've essentially taken a complex problem and turned it into a familiar one. The power of substitution is truly remarkable! By rewriting the equation, we've opened the door to a whole new set of techniques for finding the solution. It's like having a secret key that unlocks a hidden path to the answer.
Solving the Quadratic Equation: Unveiling the Roots
Now that we have a quadratic equation in terms of $u$, we can employ our favorite methods for solving quadratics. There are several approaches we can take, including factoring, completing the square, or using the quadratic formula. In this case, factoring seems like a promising approach. We're looking for two numbers that multiply to 21 and add up to -10. Those numbers are -3 and -7. So, we can factor the quadratic equation as follows: $u^2 - 10u + 21 = (u - 3)(u - 7) = 0$. This gives us two possible solutions for $u$: $u = 3$ or $u = 7$. These values of $u$ are the roots of the quadratic equation, but remember, we're ultimately interested in finding the values of $x$. We've made progress, but our journey isn't over yet. We need to reverse the substitution to find the corresponding values of $x$.
Reversing the Substitution: Finding the Values of x
We've found the values of $u$, but we need to remember the original problem: solving for $x$. To do this, we need to reverse the substitution we made earlier. Recall that we let $u = x^2$. Now we can substitute the values of $u$ we found back into this equation. If $u = 3$, then $x^2 = 3$, which means $x = \pm\sqrt3}$. Similarly, if $u = 7$, then $x^2 = 7$, which means $x = \pm\sqrt{7}$. So, we have four solutions for $x$$, $-\sqrt{3}$, $\sqrt{7}$, and $-\sqrt{7}$. These are the roots of the original quartic equation. We've successfully navigated the complexities of the equation by using the substitution method. It's like piecing together a puzzle, where each step brings us closer to the final solution. And now, we've arrived at the complete picture!
The Complete Solution Set: All Roots Revealed
We've done it! We've solved the equation $x^4 - 10x^2 + 21 = 0$ using the method of substitution. Our solutions are $x = \sqrt3}$, $x = -\sqrt{3}$, $x = \sqrt{7}$, and $x = -\sqrt{7}$. These four values are the roots of the equation, meaning they are the values of $x$ that make the equation true. We can write the solution set as, -\sqrt{3}, \sqrt{3}, \sqrt{7}}$. This set represents all the possible solutions to the equation. By using substitution, we were able to transform a seemingly difficult quartic equation into a manageable quadratic equation, which we then solved using factoring. The substitution method is a powerful tool in algebra, allowing us to simplify complex problems and find solutions that might otherwise be elusive. It's a testament to the elegance and ingenuity of mathematical techniques.
Conclusion: The Power of Substitution in Action
In this article, we've explored the method of substitution, a valuable technique for solving equations. We used the equation $x^4 - 10x^2 + 21 = 0$ as an example, demonstrating how substitution can transform a complex quartic equation into a simpler quadratic equation. By letting $u = x^2$, we were able to rewrite the equation in terms of $u$, solve for $u$, and then reverse the substitution to find the values of $x$. The solutions we found were $x = \sqrt{3}$, $x = -\sqrt{3}$, $x = \sqrt{7}$, and $x = -\sqrt{7}$. The method of substitution is a versatile tool that can be applied to a wide range of equations. It's a testament to the power of algebraic manipulation and the beauty of mathematical problem-solving. By understanding and mastering techniques like substitution, we can unlock the solutions to even the most challenging equations. So, keep exploring, keep practicing, and keep discovering the wonders of mathematics!
Fill in the blanks:
We solve $x^4-10 x^2+21=0$ by letting $u=$ x^2. We then rewrite the equation in terms of $u$ as u^2 - 10u + 21 = 0.