Solving $x^4-40x^2+144=0$ Using Substitution A Step-by-Step Guide

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Solving equations can sometimes seem like navigating a complex maze, but with the right strategies, even the most daunting problems can be broken down into manageable steps. In this comprehensive guide, we will delve into the technique of substitution to solve the equation $x^4 - 40x^2 + 144 = 0$. This method allows us to transform a seemingly complicated equation into a simpler, more familiar form, making it easier to find the solutions. We will explore each step in detail, ensuring a clear understanding of the process and the underlying principles. By the end of this guide, you will not only be able to solve this specific equation but also gain a valuable tool for tackling a wide range of similar problems.

Understanding the Power of Substitution

Before we dive into the specifics of our equation, let's first appreciate the power of substitution. Substitution is a powerful technique in algebra that involves replacing a complex expression with a single variable. This simplifies the equation, making it easier to manipulate and solve. Think of it as a translator that converts a foreign language into one you understand. In our case, the 'foreign language' is the quartic equation ($x^4 - 40x^2 + 144 = 0$), and we will 'translate' it into a quadratic equation, which we know how to solve. This method is particularly useful when dealing with equations that have a repeating pattern or structure, such as the one we are about to tackle. The ability to recognize these patterns and apply substitution is a crucial skill in mathematical problem-solving.

Step-by-Step Solution

Now, let's embark on the journey of solving the equation $x^4 - 40x^2 + 144 = 0$ using substitution. We'll break down the process into clear, concise steps, ensuring that you grasp each concept along the way. This methodical approach will not only help you solve this particular problem but also equip you with a valuable problem-solving strategy for future challenges.

1. Identify the Pattern and Make the Substitution

Our equation, $x^4 - 40x^2 + 144 = 0$, might look intimidating at first glance, but notice the pattern: the exponent of the first term ($x^4$) is twice the exponent of the second term ($x^2$). This is a key indicator that substitution can be a powerful tool here. Let's make the substitution:

$u = x^2$

This seemingly simple substitution is the cornerstone of our solution. By replacing $x^2$ with $u$, we are effectively changing the landscape of the equation, making it more accessible. This step highlights the importance of pattern recognition in mathematical problem-solving. Spotting these patterns allows us to choose the most effective strategies and simplify complex problems.

Now, let's apply this substitution to our equation. Replacing every instance of $x^2$ with $u$, we get:

$u^2 - 40u + 144 = 0$

Notice how the equation has transformed. What was once a quartic equation (degree 4) is now a quadratic equation (degree 2) in terms of $u$. This transformation is the essence of the substitution method. We've taken a complex problem and simplified it into a familiar form.

2. Solve the Quadratic Equation

The equation $u^2 - 40u + 144 = 0$ is a standard quadratic equation. We have several methods at our disposal to solve it, such as factoring, completing the square, or using the quadratic formula. For this example, let's use factoring, as it's often the quickest method when applicable.

We need to find two numbers that multiply to 144 and add up to -40. After some consideration, we can identify -4 and -36 as the numbers that satisfy these conditions:

(-4) * (-36) = 144

(-4) + (-36) = -40

Now we can rewrite the quadratic equation in factored form:

$(u - 4)(u - 36) = 0$

This factored form is incredibly useful because it allows us to directly identify the solutions for $u$. For the product of two factors to be zero, at least one of the factors must be zero. Therefore, we have two possibilities:

$u - 4 = 0$ or $u - 36 = 0$

Solving each of these equations for $u$, we get:

$u = 4$ or $u = 36$

These are the solutions for $u$, but remember, our original problem was in terms of $x$. We need to 'translate' these solutions back to the original variable.

3. Substitute Back to Find x

We've successfully solved for $u$, but our ultimate goal is to find the values of $x$ that satisfy the original equation. This is where the second part of the substitution process comes into play – substituting back. Remember our original substitution:

$u = x^2$

Now, we'll use this relationship to find the values of $x$ corresponding to each value of $u$. We have two cases to consider:

Case 1: u = 4

Substitute $u = 4$ back into the equation $u = x^2$:

$x^2 = 4$

To solve for $x$, we take the square root of both sides. Remember that when taking the square root, we need to consider both positive and negative solutions:

$x = ±

Therefore, we have two solutions for this case:

$x = 2$ or $x = -2$

Case 2: u = 36

Now, let's substitute $u = 36$ back into the equation $u = x^2$:

$x^2 = 36$

Again, we take the square root of both sides, remembering to consider both positive and negative solutions:

$x = ±

This gives us two more solutions:

$x = 6$ or $x = -6$

4. List the Solutions

We've diligently worked through the substitution process, and now we have all the solutions for the equation $x^4 - 40x^2 + 144 = 0$. Let's gather them together:

$x = -6, -2, 2, 6$

These are the four values of $x$ that satisfy the original equation. We have successfully navigated the complexity of the quartic equation by using the power of substitution.

Verification (Optional but Recommended)

As a final step, it's always a good practice to verify our solutions. This ensures that we haven't made any errors along the way. We can do this by plugging each solution back into the original equation and checking if it holds true.

Verification for x = -6

Substitute $x = -6$ into $x^4 - 40x^2 + 144 = 0$:

$(-6)^4 - 40(-6)^2 + 144 = 1296 - 40(36) + 144 = 1296 - 1440 + 144 = 0$

The equation holds true.

Verification for x = -2

Substitute $x = -2$ into $x^4 - 40x^2 + 144 = 0$:

$(-2)^4 - 40(-2)^2 + 144 = 16 - 40(4) + 144 = 16 - 160 + 144 = 0$

The equation holds true.

Verification for x = 2

Substitute $x = 2$ into $x^4 - 40x^2 + 144 = 0$:

$(2)^4 - 40(2)^2 + 144 = 16 - 40(4) + 144 = 16 - 160 + 144 = 0$

The equation holds true.

Verification for x = 6

Substitute $x = 6$ into $x^4 - 40x^2 + 144 = 0$:

$(6)^4 - 40(6)^2 + 144 = 1296 - 40(36) + 144 = 1296 - 1440 + 144 = 0$

The equation holds true.

Since all four solutions satisfy the original equation, we can confidently conclude that our solutions are correct.

Conclusion: Mastering the Art of Substitution

In this comprehensive guide, we've successfully solved the equation $x^4 - 40x^2 + 144 = 0$ using the powerful technique of substitution. We've seen how substitution can transform a complex equation into a simpler, more manageable form. By replacing $x^2$ with a new variable, $u$, we converted a quartic equation into a quadratic equation, which we could then solve using familiar methods.

The key takeaways from this guide are:

• Recognizing patterns: Identifying repeating patterns in equations is crucial for applying substitution effectively.
• Making the substitution: Choosing the right substitution is essential for simplifying the equation.
• Solving the transformed equation: Once the substitution is made, solve the resulting equation using appropriate methods.
• Substituting back: Don't forget to substitute back to find the solutions in terms of the original variable.
• Verification: Always verify your solutions to ensure accuracy.

Substitution is a versatile technique that can be applied to a wide range of equations. By mastering this method, you'll be well-equipped to tackle more complex mathematical problems. Remember, practice is key. The more you use substitution, the more comfortable and confident you'll become in applying it. Keep exploring, keep learning, and keep solving!