Unveiling Luca's Equation Error A Detailed Mathematical Explanation

#H1 Luca's Equation Error Unveiled: A Mathematical Deep Dive

Luca attempted to solve the equation $\sqrt{2x-5} = \sqrt{x-3} + 1$, but made a critical error in the process. His steps are as follows:

$\begin{aligned} \sqrt{2 x-5} & =\sqrt{x-3}+1 \\ (\sqrt{2 x-5})^2 & =(\sqrt{x-3}+1)^2 \\ 2 x-5 & =x-3+1 \\ x & =3 \end{aligned}$

Let's dissect Luca's solution to pinpoint the exact mistake and understand why it led to an incorrect result. This exploration will not only highlight the error but also reinforce the importance of careful algebraic manipulation, especially when dealing with square roots and radicals. By understanding where Luca went wrong, we can learn valuable lessons about equation-solving techniques and avoid similar pitfalls in the future.

Identifying Luca's Error: The Forgotten Cross Term

The primary error in Luca's solution lies in the incorrect squaring of the binomial $(\sqrt{x-3} + 1)^2$. When squaring a binomial of the form $(a + b)^2$, the result is $a^2 + 2ab + b^2$, not simply $a^2 + b^2$. Luca seems to have missed the crucial middle term, the cross term, which is $2ab$. This oversight is a common mistake in algebra and can lead to significant errors in solving equations. In this specific case, failing to account for the cross term during the squaring process is the root cause of the incorrect solution. The implications of this error cascade through the rest of the steps, ultimately leading to an inaccurate value for $x$.

When Luca squared both sides of the equation, he correctly squared the left side, $(\sqrt{2x-5})^2$, which simplifies to $2x - 5$. However, on the right side, he incorrectly calculated $(\sqrt{x-3} + 1)^2$ as $x - 3 + 1$. The correct expansion should be: $(\sqrt{x-3} + 1)^2 = (\sqrt{x-3})^2 + 2(\sqrt{x-3})(1) + 1^2 = x - 3 + 2\sqrt{x-3} + 1$. The missing term is $2\sqrt{x-3}$, which is the crucial cross term that Luca overlooked. This omission drastically changes the equation and leads to an erroneous result. Understanding the correct expansion of binomial squares is fundamental to solving algebraic equations accurately. The correct application of the binomial square formula is not just a matter of algebraic technique; it’s a matter of mathematical precision.

To further emphasize the importance of the cross term, let's consider a numerical example. Suppose we have the expression $(2 + 3)^2$. According to the correct formula, this should be $2^2 + 2(2)(3) + 3^2 = 4 + 12 + 9 = 25$. If we mistakenly ignore the cross term, we would calculate it as $2^2 + 3^2 = 4 + 9 = 13$, which is significantly different. This simple example illustrates how crucial the cross term is in binomial expansion. In the context of Luca's equation, the term $2\sqrt{x-3}$ plays a similar role, and omitting it fundamentally alters the equation's structure and solution. Therefore, recognizing and correctly applying the binomial square formula is essential for accuracy in algebraic manipulations and equation solving.

Correcting the Solution: A Step-by-Step Approach

To rectify Luca's error, we need to square the binomial $(\sqrt{x-3} + 1)$ correctly. Here's the step-by-step solution:

1. Start with the original equation: $\sqrt{2x-5} = \sqrt{x-3} + 1$
2. Square both sides, paying attention to the cross term: $(\sqrt{2x-5})^2 = (\sqrt{x-3} + 1)^2$ $2x - 5 = (x - 3) + 2\sqrt{x-3} + 1$
3. Simplify the equation: $2x - 5 = x - 2 + 2\sqrt{x-3}$
4. Isolate the square root term: $x - 3 = 2\sqrt{x-3}$
5. Square both sides again: $(x - 3)^2 = (2\sqrt{x-3})^2$ $x^2 - 6x + 9 = 4(x - 3)$
6. Further simplification and rearrangement: $x^2 - 6x + 9 = 4x - 12$ $x^2 - 10x + 21 = 0$
7. Factor the quadratic equation: $(x - 7)(x - 3) = 0$
8. Solve for x: $x = 7$ or $x = 3$

Verifying the Solutions: Checking for Extraneous Roots

It is crucial to verify the solutions obtained, particularly when dealing with square root equations. Squaring both sides of an equation can sometimes introduce extraneous roots, which are solutions that satisfy the transformed equation but not the original equation. Therefore, we need to substitute each solution back into the original equation to check its validity. This process of verification is a vital step in solving equations involving radicals. By checking our solutions, we ensure that they are not just mathematical artifacts but genuine solutions to the problem at hand. This practice reinforces the integrity of the solution process and prevents us from accepting false answers.

Let's start by checking $x = 7$:

$\sqrt{2(7) - 5} = \sqrt{7 - 3} + 1$ $\sqrt{14 - 5} = \sqrt{4} + 1$ $\sqrt{9} = 2 + 1$ $3 = 3$

This confirms that $x = 7$ is a valid solution. The substitution shows that the equation holds true when $x$ is 7, indicating that it is a genuine solution and not an extraneous root. This step underscores the importance of not just finding a potential solution but also verifying its authenticity. In the realm of mathematical problem-solving, verification is the ultimate test of a solution's validity.

Now, let's check $x = 3$:

$\sqrt{2(3) - 5} = \sqrt{3 - 3} + 1$ $\sqrt{6 - 5} = \sqrt{0} + 1$ $\sqrt{1} = 0 + 1$ $1 = 1$

Although $x = 3$ satisfies the original equation, it's essential to recognize that it arose from a step where we squared both sides, which is a potential source of extraneous solutions. While it does work in this case, it doesn't negate the importance of the verification step. In other similar problems, such a check might reveal that $x=3$ is indeed an extraneous root. Thus, the solution $x = 3$ is also a valid solution.

The Importance of Checking Solutions

The process of checking solutions, especially in equations involving radicals or rational expressions, cannot be overemphasized. Squaring both sides, as done in this problem, or multiplying both sides by an expression containing a variable, can introduce extraneous solutions. These solutions are mathematically correct for the transformed equation but do not satisfy the original equation. Therefore, checking each potential solution in the original equation is a critical step in ensuring the accuracy of the final answer. This practice is not just a formality but a necessary safeguard against accepting incorrect solutions. It highlights the importance of a thorough and meticulous approach to mathematical problem-solving, where every step is verified and validated.

Extraneous solutions arise because the squaring operation, for instance, can make two unequal quantities appear equal. For example, if we have $-2$ and $2$, which are not equal, squaring both gives us $4$, making them appear equal. This is precisely why we need to check solutions in the original equation. The act of checking helps us filter out these artificial solutions, leaving only the genuine ones. This concept is fundamental in algebra and is particularly relevant in advanced mathematical contexts. Therefore, students and practitioners of mathematics must cultivate the habit of verifying their solutions, especially in situations where transformations like squaring or multiplying by variable expressions are involved.

Conclusion: Precision and Verification in Equation Solving

In summary, Luca's error was in the incorrect squaring of the binomial $(\sqrt{x-3} + 1)$. He missed the cross term, leading to an incorrect simplification and ultimately a flawed solution process. The correct solution involves careful expansion, isolation of the square root, and solving the resulting quadratic equation. Furthermore, the crucial step of verifying the solutions highlights the importance of checking for extraneous roots. This problem serves as a valuable lesson in the necessity of precision in algebraic manipulation and the critical role of verification in ensuring the accuracy of mathematical solutions. The process of solving equations is not just about finding a numerical answer; it's about ensuring that the answer is valid and consistent with the original problem. This understanding is vital for success in mathematics and related fields.

By meticulously applying algebraic principles and rigorously verifying solutions, we can navigate the complexities of equation-solving with confidence and accuracy. The lessons learned from Luca's error extend beyond this specific problem, reinforcing the importance of a thorough and methodical approach to mathematics. Each step in the solution process must be carefully considered, and potential pitfalls, such as extraneous solutions, must be actively addressed. This commitment to precision and verification is the hallmark of a skilled problem-solver and is essential for achieving reliable and meaningful results in mathematics.