Transforming Equations: Quadratic Substitution For $4x^4-21x^2+20=0$

Dec 1, 2025 by Alex Johnson 69 views

In mathematics, especially when dealing with polynomial equations, strategic substitutions can significantly simplify the problem-solving process. One common technique involves transforming a higher-degree polynomial equation into a quadratic equation, which we know how to solve using various methods such as factoring, completing the square, or the quadratic formula. This article delves into how to identify the correct substitution to rewrite the equation $4x^4 - 21x^2 + 20 = 0$ into a quadratic form. This skill is crucial not only for solving specific equations but also for developing a deeper understanding of algebraic manipulations and problem-solving strategies. So, let's break down the process and uncover the best approach to tackle this type of equation.

Understanding Quadratic Form

To understand the appropriate substitution, it's crucial first to recognize what constitutes a quadratic form. A quadratic equation is generally expressed as $ax^2 + bx + c = 0$ , where a, b, and c are constants, and x is the variable. The key characteristic of a quadratic equation is the highest power of the variable being 2. When we talk about an equation being in quadratic form, we mean that it can be rewritten to resemble this structure, even if the original equation has higher powers. For instance, an equation like $ax^4 + bx^2 + c = 0$ can be transformed into a quadratic equation through a simple substitution. The goal is to find a substitution that reduces the higher powers into a squared term, thereby making the equation solvable using quadratic techniques. Recognizing the underlying structure is the first step in choosing the right substitution strategy. By carefully analyzing the exponents and coefficients, we can make an informed decision that simplifies the equation and brings us closer to a solution. This foundational understanding is essential for mastering more complex algebraic manipulations and problem-solving scenarios.

Identifying the Quadratic Form in $4x^4 - 21x^2 + 20 = 0$

When we look at the equation $4x^4 - 21x^2 + 20 = 0$ , it might not immediately appear to be a quadratic equation. However, a closer inspection reveals that it shares a structural similarity with the standard quadratic form. Notice that the powers of x are 4 and 2, and there's also a constant term. This arrangement suggests that we can rewrite the equation in a quadratic form by making a suitable substitution. The critical observation here is the relationship between the exponents: the exponent 4 is twice the exponent 2. This hints at the possibility of substituting a new variable for $x^2$ , which, when squared, will give us $x^4$ . By doing so, we can transform the equation into a more manageable quadratic equation in terms of the new variable. Recognizing these patterns and relationships within the equation is essential for identifying the appropriate substitution strategy. This step-by-step analysis not only simplifies the problem at hand but also enhances our ability to tackle similar algebraic challenges in the future. Mastering this technique is a valuable asset in advanced mathematics and problem-solving.

Choosing the Correct Substitution

To choose the correct substitution for transforming $4x^4 - 21x^2 + 20 = 0$ into a quadratic equation, we need to identify a variable that, when substituted, will result in the familiar $au^2 + bu + c = 0$ form. The key lies in recognizing the relationship between the terms $x^4$ and $x^2$ . As we observed earlier, $x^4$ is the square of $x^2$ . This suggests that if we let a new variable, say u, be equal to $x^2$ , then $u^2$ would be equal to $(x^2)^2$ , which simplifies to $x^4$ . This substitution elegantly transforms the original equation into a quadratic equation in terms of u. Specifically, if we substitute $u = x^2$ , the equation $4x^4 - 21x^2 + 20 = 0$ becomes $4u^2 - 21u + 20 = 0$ , which is clearly a quadratic equation. This strategic choice of substitution allows us to apply well-known methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. Therefore, understanding the structure of the equation and the relationship between its terms is crucial for selecting the most effective substitution method. This approach not only simplifies the equation but also paves the way for a straightforward solution.

Evaluating the Options

When evaluating the options for substitution in the equation $4x^4 - 21x^2 + 20 = 0$ , we consider each choice to determine which one effectively transforms the equation into a quadratic form. Let's examine the given options:

A. $u = x^2$ : This substitution, as discussed earlier, is the most appropriate choice. By replacing $x^2$ with u, we directly convert the equation into the quadratic form $4u^2 - 21u + 20 = 0$ . This substitution aligns perfectly with the structure of the equation, making it easier to solve using standard quadratic techniques.
B. $u = 2x^2$ : While this substitution might seem like a viable option, it introduces an additional coefficient that complicates the resulting quadratic equation. If we substitute $u = 2x^2$ , then $x^2 = rac{u}{2}$ , and $x^4 = ( rac{u}{2})^2 = rac{u^2}{4}$ . Plugging these into the original equation gives us $4( rac{u^2}{4}) - 21( rac{u}{2}) + 20 = 0$ , which simplifies to $u^2 - rac{21}{2}u + 20 = 0$ . Although this is still a quadratic equation, the fractional coefficient makes it less straightforward to solve compared to option A.
C. $u = x^4$ : This substitution would transform the equation into a linear equation in terms of u, not a quadratic equation. If we substitute $u = x^4$ , then $x^2 = extstyle rac{u^ rac{1}{2}}$ and the equation becomes $4u - 21 extstyle rac{u^ rac{1}{2}}{} + 20 = 0$ . This equation is not quadratic and is more complex to solve.
D. $u = 4x^4$ : Similar to option C, this substitution does not result in a quadratic equation. Substituting $u = 4x^4$ means $x^4 = rac{u}{4}$ , and $x^2 = extstyle rac{u^ rac{1}{2}}{2}$ . The resulting equation, $4( rac{u}{4}) - 21 extstyle rac{u^ rac{1}{2}}{2} + 20 = 0$ , simplifies to $u - rac{21}{2} extstyle{u^ rac{1}{2}} + 20 = 0$ , which is also not a quadratic equation and is more difficult to solve.

The Correct Choice: $u = x^2$

Based on the analysis of each option, it is clear that the correct substitution to rewrite the equation $4x^4 - 21x^2 + 20 = 0$ as a quadratic equation is A. $u = x^2$ . This substitution directly transforms the original equation into $4u^2 - 21u + 20 = 0$ , which is a standard quadratic equation. The simplicity and effectiveness of this substitution make it the most logical choice for solving the given equation. By recognizing the structural relationship between the terms $x^4$ and $x^2$ , we can efficiently reduce the equation to a solvable quadratic form. This approach highlights the importance of strategic substitution in simplifying complex algebraic problems. Mastering this technique not only helps in solving specific equations but also enhances overall problem-solving skills in mathematics.

Solving the Quadratic Equation

After solving the quadratic equation, the initial substitution $u = x^2$ has transformed the original equation $4x^4 - 21x^2 + 20 = 0$ into the quadratic equation $4u^2 - 21u + 20 = 0$ . Now, we can solve this quadratic equation for u using methods such as factoring, completing the square, or the quadratic formula. In this case, factoring is a straightforward approach. We look for two numbers that multiply to $4 imes 20 = 80$ and add up to -21. These numbers are -16 and -5. Thus, we can rewrite the middle term as $-16u - 5u$ , and the equation becomes:

$4u^2 - 16u - 5u + 20 = 0$

Next, we factor by grouping:

$4u(u - 4) - 5(u - 4) = 0$

This simplifies to:

$(4u - 5)(u - 4) = 0$

Setting each factor equal to zero gives us the solutions for u:

$4u - 5 = 0 ext{ or } u - 4 = 0$

Solving for u yields:

$u = rac{5}{4} ext{ or } u = 4$

These are the solutions for the substituted variable u. However, we need to find the solutions for the original variable x. To do this, we substitute back $x^2$ for u.

Substituting Back to Find x

To substitute back to find x, we recall our original substitution, $u = x^2$ . We now have two values for u: $rac{5}{4}$ and 4. We will substitute each of these values back into the equation $u = x^2$ to find the corresponding values of x. For $u = rac{5}{4}$ , we have:

$x^2 = rac{5}{4}$

Taking the square root of both sides gives us:

$x = extstyle rac{\sqrt{5}}{2}$ or $x = - extstyle rac{\sqrt{5}}{2}$

For $u = 4$ , we have:

$x^2 = 4$

Taking the square root of both sides gives us:

$x = 2 ext{ or } x = -2$

Therefore, the solutions for the original equation $4x^4 - 21x^2 + 20 = 0$ are $x = extstyle rac{\sqrt{5}}{2}$ , $x = - extstyle rac{\sqrt{5}}{2}$ , $x = 2$ , and $x = -2$ . By substituting back, we successfully found the values of the original variable x that satisfy the equation. This step is crucial in solving equations using substitution methods, as it ensures we obtain the solutions in terms of the original variables. This process demonstrates the complete method of solving such equations, from the initial substitution to finding the final solutions.

Conclusion

In conclusion, the appropriate substitution to rewrite the equation $4x^4 - 21x^2 + 20 = 0$ as a quadratic equation is A. $u = x^2$ . This substitution transforms the equation into a manageable quadratic form, which can then be solved using standard techniques like factoring. By understanding the structure of the equation and recognizing the relationship between the terms, we can effectively simplify complex algebraic problems. The process involves substituting $u = x^2$ , solving the resulting quadratic equation for u, and then substituting back to find the solutions for x. This method not only solves the specific equation at hand but also enhances problem-solving skills in mathematics more broadly. Strategic substitution is a powerful tool in algebra, and mastering this technique is invaluable for tackling higher-level mathematical challenges.

For further exploration of algebraic techniques and equation solving, visit Khan Academy's Algebra Section. This resource offers a wealth of information and practice problems to deepen your understanding.