Solving Quartic Equations: Finding The Right Substitution

Hey math enthusiasts! Today, we're diving into a cool problem: rewriting a quartic equation (an equation with a term raised to the fourth power) as a quadratic equation (an equation with a term raised to the second power). Specifically, we'll tackle the equation $4x^4 - 21x^2 + 20 = 0$. The key to solving this lies in a clever substitution. So, what substitution should be used to rewrite $4x^4 - 21x^2 + 20 = 0$ as a quadratic equation? Let's break it down and find the right answer. This process simplifies the equation and makes it easier to solve using methods we already know.

Understanding the Problem: Quartic vs. Quadratic

Before we jump into the options, let's make sure we're all on the same page. The equation $4x^4 - 21x^2 + 20 = 0$ is a quartic equation. This means the highest power of the variable (in this case, x) is 4. Solving quartic equations directly can be tricky, often involving complex formulas. However, we can make our lives much easier by transforming it into a quadratic equation. Quadratic equations have the form $ax^2 + bx + c = 0$, where a, b, and c are constants. We know a lot about solving these – like factoring, using the quadratic formula, or completing the square. The goal is to manipulate our original quartic equation into this familiar quadratic form.

Now, why is this helpful? Because quadratic equations are generally easier to solve. We have established methods for finding their roots, making the overall process less cumbersome. By carefully selecting a substitution, we can reduce the complexity of the equation and make it much more manageable. Think of it as a strategic move to simplify the puzzle and uncover its solution efficiently. This technique is a fundamental tool in algebra, streamlining the solution process and avoiding unnecessary complexity.

Analyzing the Answer Choices

Now, let's look at the answer choices provided and see which substitution works best. We need to choose the substitution that, when applied to the original equation, will result in an equation that looks like a quadratic. Here are the options:

A. $u = 2x^2$ B. $u = x^2$ C. $u = x^4$ D. $u = 4x^4$

To figure this out, we need to think about how each substitution would change the equation. Let's start with option B, $u = x^2$. If we substitute $u$ for $x^2$, then $x^4$ becomes $(x^2)^2$, which is $u^2$. Let's plug it in and see what happens.

The Correct Substitution: $u = x^2$

Let's apply the substitution $u = x^2$ to the original equation $4x^4 - 21x^2 + 20 = 0$. Remember, we want to transform this into a quadratic equation, which has the general form $au^2 + bu + c = 0$. If we replace every instance of $x^2$ with u, here’s what we get:

• $x^4$ becomes $(x^2)^2$, which is $u^2$
• $x^2$ becomes $u$

So, our equation $4x^4 - 21x^2 + 20 = 0$ transforms into $4u^2 - 21u + 20 = 0$. Now, doesn't that look familiar? It’s a standard quadratic equation in terms of u! This means we can solve it using the quadratic formula, factoring, or any other method we know for solving quadratics. Once we find the values of u, we can easily find the values of x by going back to our substitution, $u = x^2$. This is the magic of substitution – simplifying a complex equation into a form we can readily solve. This approach provides a clear path to the solution.

Applying this substitution gives us an equation that we can solve using standard quadratic methods. This strategic move unlocks the solution by converting a complex equation into a more approachable format.

Why Other Options Don't Work

Let's quickly go through why the other answer choices don't work as well.

• A. $u = 2x^2$: If we substitute $u = 2x^2$, then $x^2 = u/2$ and $x^4 = (u/2)^2 = u^2/4$. Substituting these into the original equation would result in something more complex, not a straightforward quadratic. The presence of the coefficient '2' complicates the transformation and does not directly yield the desired quadratic form.
• C. $u = x^4$: If we substitute $u = x^4$, then the equation becomes 4u - 21rac{\sqrt{u}}{} + 20 = 0. This substitution doesn’t lead to a quadratic equation. It introduces a square root term, making the equation more complex instead of simpler. This substitution does not effectively reduce the degree of the equation, which is crucial for simplifying the solution process.
• D. $u = 4x^4$: If we substitute $u = 4x^4$, then $x^2 = \sqrt{u/4}$. This substitution does not readily transform the equation into a quadratic form. While it simplifies the first term, it complicates the second term, resulting in an equation that is not easier to solve as a quadratic. Therefore, option D does not provide the direct simplification we seek.

Conclusion: The Power of Substitution

So, the correct answer is B. $u = x^2$. By making this substitution, we successfully transformed the quartic equation into a quadratic equation, which we can solve using familiar methods. This is a common and powerful technique in algebra, allowing us to simplify complex equations and find solutions efficiently.

This method demonstrates a fundamental concept: strategic manipulation can significantly reduce the complexity of mathematical problems, making them more accessible and solvable. By recognizing the patterns in the equations and choosing the right substitutions, we gain a valuable tool for tackling a wide range of algebraic challenges. Using this approach, you can effectively navigate and solve complex equations.

Keep practicing, and you'll become a pro at these types of problems. Happy solving, guys!