Common Mistakes When Taking the Square Root of Both Sides of an Equation
Many students and even professional mathematicians sometimes fall into the trap of incorrectly assuming that the only solution to an equation is the positive square root. However, it is crucial to remember, as we will explain, that there can be both positive and negative roots, depending on the context. This article aims to clear up any confusion surrounding the square rooting of both sides of an equation by exploring a common mistake and its implications.
Understanding the Square Root Function
In mathematics, the square root of a number 'x' is a value that, when multiplied by itself, gives the original number 'x'. This is often denoted as sqrt{x}. By definition, the principal square root is the positive non-negative value that satisfies sqrt{x}^2 x. However, it's important to note that every non-zero real number (except zero) has two square roots: a positive and a negative one. For instance, the equation x^2 16 has solutions x 4 and x -4, since both (4)^2 16 and (-4)^2 16.
A Common Misunderstanding: The "Proof" That 4 5
Let's look at a well-known fallacy to illustrate this issue. Here, we will take a closer look at a purported "proof" that equates 4 to 5, which has often been used in mathematics teaching to demonstrate the importance of careful algebraic manipulation. The fallacy arises from a common mistake when taking the square root of both sides of an equation.
Start with the true statement:
-20 -20
Add 16 - 36 to both sides:
-20 16 - 36 -20 16 - 36
-20 (4^2 - 36) -20 (4^2 - 36)
Add 25 - 45 to both sides:
-20 16 - 36 25 - 45 -20 16 - 36 25 - 45
-20 (4^2 - 36) (5^2 - 45) -20 (4^2 - 36) (5^2 - 45)
Rearrange to match the form of a square of two terms:
-20 4^2 - 36 5^2 - 45 -20 4^2 - 36 5^2 - 45
(4^2 - 2 cdot 4 cdot frac{9}{2} (frac{9}{2})^2) - (5^2 - 2 cdot 5 cdot frac{9}{2} (frac{9}{2})^2) 0
Factorize the left side:
(4 - frac{9}{2})^2 - (5 - frac{9}{2})^2 0
((4 - frac{9}{2}) - (5 - frac{9}{2}))((4 - frac{9}{2}) (5 - frac{9}{2})) 0
(4 - frac{9}{2} - 5 frac{9}{2})(4 - frac{9}{2} 5 - frac{9}{2}) 0
(4 - 5)(9 - 4.5) 0
(-1)(4.5) 0
-4.5 0
Clearly, the last line is false. The mistake in this "proof" is in the step where the square root of both sides of an equation is taken. Specifically, when we square both sides, we are eliminating the distinction between positive and negative square roots. This leads to the conclusion that 4 - frac{9}{2} 5 - frac{9}{2}, which simplifies to -0.5 0.5, which is obviously false.
Key Takeaways and Precautions
Always remember that when you take the square root of both sides of an equation, you must consider both the positive and the negative roots.
Be wary of cancelling or rearranging terms without checking the domain of the equation, as this can introduce extraneous solutions.
When dealing with equations involving square roots, always verify your solutions by substituting them back into the original equation.
Be cautious of algebraic manipulations that lead to squared terms, as they often obfuscate the issue at hand.
Understanding these principles and practicing careful algebraic manipulations can prevent many common mistakes in solving equations. This article aims to provide a clearer understanding of the pitfalls that can arise from seemingly innocent algebraic steps and to strengthen your ability to solve equations accurately.