Solving X⁴ - 7x² + 6 = 0 Find All Real Solutions

Finding the real solutions for polynomial equations is a fundamental problem in algebra. This article delves into solving a specific quartic equation, x⁴ - 7x² + 6 = 0. We'll explore the techniques used to find all real values of x that satisfy this equation, providing a step-by-step guide that's both comprehensive and easy to understand. Our main keywords are quartic equation, real solutions, and polynomial equation, which will be strategically used throughout the article to ensure clarity and SEO optimization.

Understanding the Problem: Quartic Equations and Real Solutions

Before diving into the solution, let's first understand the problem. A quartic equation is a polynomial equation of the fourth degree, meaning the highest power of the variable x is 4. The general form of a quartic equation is ax⁴ + bx³ + cx² + dx + e = 0, where a, b, c, d, and e are constants. Real solutions, in the context of polynomial equations, are the values of x that are real numbers and make the equation true. In other words, when you substitute these values for x in the equation, the left-hand side equals the right-hand side (which is zero in this case).

Solving quartic equations can be challenging, but this particular equation has a special form that makes it easier to tackle. Notice that the equation x⁴ - 7x² + 6 = 0 only contains even powers of x. This suggests a substitution method, which we'll explore in the next section. This polynomial equation is a quadratic in disguise, making it more manageable than a general quartic equation. The ability to recognize patterns like this is crucial in solving mathematical problems efficiently. Furthermore, understanding the nature of real solutions is essential for interpreting the results. We are looking for values on the number line that, when raised to the fourth power, squared, and combined as in the equation, yield zero. This geometric interpretation can sometimes provide additional insights into the solutions.

The Substitution Method: Transforming the Quartic into a Quadratic

The key to solving x⁴ - 7x² + 6 = 0 lies in recognizing its quadratic-like structure. To make this explicit, we introduce a substitution: Let y = x². This substitution transforms the quartic equation into a quadratic equation in y. Replacing x² with y and x⁴ with (x²)² = y², the equation becomes:

y² - 7y + 6 = 0

This is a quadratic equation, which we can solve using several methods, such as factoring, completing the square, or the quadratic formula. Factoring is often the quickest method when applicable, and in this case, the quadratic expression factors nicely. We need to find two numbers that multiply to 6 and add up to -7. These numbers are -1 and -6. Therefore, we can factor the quadratic equation as follows:

(y - 1)(y - 6) = 0

This equation is satisfied if either y - 1 = 0 or y - 6 = 0. Solving for y, we get two possible solutions:

y = 1 or y = 6

The substitution method is a powerful technique for simplifying complex equations. By introducing a new variable that represents a part of the original expression, we can transform the equation into a more manageable form. This is a common strategy in mathematics, applicable to various types of equations, not just polynomials. The ability to identify suitable substitutions is a valuable skill. In this case, recognizing that the quartic equation had a quadratic structure was the crucial insight. The new polynomial equation in y is significantly easier to solve. However, we must remember that we are not ultimately interested in the values of y, but in the values of x. The next step involves reversing the substitution to find the real solutions for x.

Reversing the Substitution: Finding the Values of x

Now that we have the solutions for y, we need to reverse the substitution y = x² to find the corresponding values of x. We have two equations to solve:

1. x² = 1
2. x² = 6

For the first equation, x² = 1, we take the square root of both sides. Remember to consider both the positive and negative square roots, as both 1² and (-1)² equal 1. Therefore, the solutions are:

x = 1 or x = -1

For the second equation, x² = 6, we again take the square root of both sides, considering both positive and negative roots. This gives us:

x = √6 or x = -√6

So, we have found four potential solutions for the original quartic equation: 1, -1, √6, and -√6. Reversing the substitution is a crucial step in the process. It connects the solutions of the transformed equation back to the original variable. This step often involves taking roots, as in this case, which is why it's essential to remember to consider both positive and negative roots. Each value of y leads to two potential values of x, reflecting the fact that the original equation is a quartic equation, and thus can have up to four real solutions. These solutions are the real solutions to the polynomial equation and must satisfy the original equation.

Verification: Checking the Solutions

It's always a good practice to verify the solutions we've found to ensure they satisfy the original equation. This helps catch any potential errors made during the solving process. Let's substitute each solution back into the equation x⁴ - 7x² + 6 = 0:

1. For x = 1: 1⁴ - 7(1²) + 6 = 1 - 7 + 6 = 0. This solution is valid.
2. For x = -1: (-1)⁴ - 7((-1)²) + 6 = 1 - 7 + 6 = 0. This solution is also valid.
3. For x = √6: (√6)⁴ - 7((√6)²) + 6 = 36 - 7(6) + 6 = 36 - 42 + 6 = 0. This solution is valid.
4. For x = -√6: (-√6)⁴ - 7((-√6)²) + 6 = 36 - 7(6) + 6 = 36 - 42 + 6 = 0. This solution is also valid.

All four solutions satisfy the original equation, confirming that we have found all the real solutions. Verification is a critical step in solving any equation. It provides a check on the correctness of the solution process and helps avoid errors. By substituting the potential solutions back into the original equation, we can confirm whether they truly make the equation true. This is particularly important for polynomial equations, where the complexity of the equation can sometimes lead to extraneous solutions. In this case, all the solutions we found passed the verification, giving us confidence that we have accurately solved the quartic equation. The verification step reinforces our understanding of the real solutions and ensures the accuracy of the answer.

Final Answer: Listing the Real Solutions

Therefore, the real solutions to the equation x⁴ - 7x² + 6 = 0 are:

x = -√6, -1, 1, √6

We have successfully found all the real solutions to the given quartic equation. The process involved recognizing the quadratic-like structure, using substitution to simplify the equation, solving the resulting quadratic equation, reversing the substitution, and verifying the solutions. This systematic approach can be applied to other polynomial equations as well. The final answer presents the complete set of real solutions for the given quartic equation. This concise list clearly communicates the outcome of our problem-solving process. The journey to finding these solutions involved several key steps, from recognizing the special structure of the polynomial equation to employing the substitution method and verifying the results. Each step contributed to the final answer and highlights the importance of a structured and methodical approach to solving mathematical problems.