Rewriting X^8 - 3x^4 + 2 = 0 As A Quadratic Equation Using Substitution

In the realm of mathematics, encountering equations that appear complex at first glance is a common challenge. However, with the right techniques, these seemingly daunting problems can often be simplified and solved elegantly. One such technique is substitution, which involves replacing a complex expression with a simpler variable to transform the equation into a more manageable form. This approach is particularly useful when dealing with equations that have a structure resembling a quadratic equation, even if they aren't quadratic in their original form. In this article, we will delve into the process of using substitution to rewrite the equation $x^8 - 3x^4 + 2 = 0$ as a quadratic equation, making it easier to solve. We'll explore the underlying principles, discuss the options for substitution, and ultimately determine the correct substitution that unlocks the quadratic nature of the equation.

Understanding the Power of Substitution

Before diving into the specific equation, let's take a moment to appreciate the power of substitution. Substitution is a fundamental technique in algebra and calculus, allowing us to simplify complex expressions and equations by introducing new variables. The core idea behind substitution is to identify a recurring expression within the equation and replace it with a single variable, thereby reducing the complexity of the equation. This often transforms the equation into a more familiar form, such as a quadratic equation, which we know how to solve using established methods.

For instance, consider an equation like $(x^2 + 1)^2 + 3(x^2 + 1) + 2 = 0$. This equation looks complicated at first, but we can observe that the expression $x^2 + 1$ appears multiple times. By substituting $u = x^2 + 1$, we transform the equation into $u^2 + 3u + 2 = 0$, which is a simple quadratic equation in $u$. Solving for $u$ and then substituting back to find $x$ becomes a straightforward process. This example highlights how substitution can significantly simplify the problem-solving process.

The key to successful substitution lies in identifying the appropriate expression to substitute. This often involves looking for patterns and recurring terms within the equation. Once the right substitution is made, the equation transforms into a more manageable form, making it easier to apply standard solution techniques.

Identifying the Quadratic Form

Our goal is to rewrite the equation $x^8 - 3x^4 + 2 = 0$ as a quadratic equation using substitution. To do this, we must first recognize the underlying quadratic form within the equation. A quadratic equation is generally expressed in the form $au^2 + bu + c = 0$, where $a$, $b$, and $c$ are constants, and $u$ is the variable. Our task is to find a substitution that transforms the given equation into this form.

Looking at the equation $x^8 - 3x^4 + 2 = 0$, we can observe a structural similarity to a quadratic equation. Notice that the exponents of $x$ are 8 and 4, and 8 is twice 4. This suggests that we might be able to rewrite the equation in a quadratic form by choosing an appropriate substitution. The equation can be seen as a polynomial in $x^4$. If we let $u$ equal some power of $x$, we want the equation to become a quadratic equation in $u$.

The crucial step is to identify the correct expression to substitute so that the equation takes the quadratic form. We need to find a variable $u$ such that the terms in the equation can be expressed as multiples of $u^2$, $u$, and a constant. Let's examine the given options and see which one fits this requirement.

Evaluating the Substitution Options

We are given four options for substitution:

A. $u = x^2$ B. $u = x^4$ C. $u = x^8$ D. $u = x^{16}$

Let's analyze each option to determine which one will successfully transform the equation $x^8 - 3x^4 + 2 = 0$ into a quadratic equation.

Option A: $u = x^2$

If we substitute $u = x^2$, then $u^2 = (x^2)^2 = x^4$. However, we also have a term $x^8$ in the original equation. If $u = x^2$, then $x^8 = (x^2)^4 = u^4$. Substituting into the original equation, we get $u^4 - 3u^2 + 2 = 0$. This equation is not a quadratic equation; it's a quartic equation in $u$. Therefore, option A is not the correct substitution.

Option B: $u = x^4$

If we substitute $u = x^4$, then $u^2 = (x^4)^2 = x^8$. Substituting into the original equation, we get $x^8 - 3x^4 + 2 = 0$ becomes $u^2 - 3u + 2 = 0$. This equation is a quadratic equation in $u$. This substitution successfully transforms the original equation into a quadratic form, making it easier to solve. Therefore, option B appears to be the correct substitution.

Option C: $u = x^8$

If we substitute $u = x^8$, the equation becomes $u - 3x^4 + 2 = 0$. We still have the $x^4$ term, which we need to express in terms of $u$. Since $u = x^8$, then $x^4 = u^{1/2}$. Substituting this into the equation gives us $u - 3u^{1/2} + 2 = 0$. This equation is not a quadratic equation; it involves a square root term. Therefore, option C is not the correct substitution.

Option D: $u = x^{16}$

If we substitute $u = x^{16}$, we need to express $x^8$ and $x^4$ in terms of $u$. Since $u = x^{16}$, then $x^8 = u^{1/2}$ and $x^4 = u^{1/4}$. Substituting these into the original equation gives us $u^{1/2} - 3u^{1/4} + 2 = 0$. This equation is not a quadratic equation; it involves fractional exponents. Therefore, option D is not the correct substitution.

The Correct Substitution and the Transformed Equation

Based on our analysis, the correct substitution to rewrite the equation $x^8 - 3x^4 + 2 = 0$ as a quadratic equation is:

B. $u = x^4$

When we substitute $u = x^4$ into the equation, we obtain:

$u^2 - 3u + 2 = 0$

This is a quadratic equation in the variable $u$. Now, we can solve this quadratic equation using various methods, such as factoring, completing the square, or the quadratic formula.

Solving the Quadratic Equation

The quadratic equation $u^2 - 3u + 2 = 0$ can be solved by factoring. We look for two numbers that multiply to 2 and add up to -3. These numbers are -1 and -2. Therefore, we can factor the quadratic equation as follows:

$(u - 1)(u - 2) = 0$

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

$u - 1 = 0$ or $u - 2 = 0$

$u = 1$ or $u = 2$

Substituting Back to Find x

Now that we have the solutions for $u$, we need to substitute back to find the solutions for $x$. Recall that we made the substitution $u = x^4$. Therefore, we have two equations to solve:

1. $x^4 = 1$
2. $x^4 = 2$

Solving $x^4 = 1$

To solve $x^4 = 1$, we take the fourth root of both sides:

$x = oots[4]{1}$

The fourth roots of 1 are 1, -1, $i$, and $-i$, where $i$ is the imaginary unit ($\sqrt{-1}$). Therefore, the solutions for $x$ in this case are:

$x = 1, -1, i, -i$

Solving $x^4 = 2$

To solve $x^4 = 2$, we take the fourth root of both sides:

$x = oots[4]{2}$

This equation has four complex solutions. In the real number system, the solutions are $x = oots[4]{2}$ and $x = - oots[4]{2}$. The other two solutions are complex numbers.

Conclusion

In conclusion, the correct substitution to rewrite the equation $x^8 - 3x^4 + 2 = 0$ as a quadratic equation is B. $u = x^4$. This substitution transforms the equation into $u^2 - 3u + 2 = 0$, which is a quadratic equation in $u$. Solving for $u$ and then substituting back allows us to find the solutions for $x$. This example demonstrates the power of substitution as a technique for simplifying complex equations and making them more accessible to solve. By recognizing patterns and choosing appropriate substitutions, we can transform seemingly challenging problems into manageable ones.

The process of substitution is a powerful tool in mathematics, allowing us to simplify equations and solve them more easily. By understanding the underlying principles and practicing identifying appropriate substitutions, we can enhance our problem-solving skills and tackle a wider range of mathematical challenges. The ability to transform equations into familiar forms, such as quadratic equations, is a valuable asset in any mathematical endeavor. Remember to always check your solutions by substituting them back into the original equation to ensure their validity.