Determine where the following reasoning is flawed:
[tex]$7 \sqrt{(x-1)^2 + 2x - 1} = 7 \sqrt{(x^2 - 2x + 1) + 2x - 1} = 7 \sqrt{x^2} = x$[/tex]
A. In the last root, you can choose the value of the unknown so that its square is negative.
B. The last equality is stating that the square root of a positive number can be a negative number.
C. Within the expressions found in the first two square roots, there are negative terms.
D. In the first equality, the expansion of the perfect square trinomial inside the root has a negative coefficient.
Answer (2)
The reasoning incorrectly states that 7 x 2 = x .
The correct statement is 7 x 2 = 7∣ x ∣ .
The error lies in assuming that the square root of a number squared is always the number itself, without considering the absolute value.
Therefore, the correct answer is that the last equality is affirming that the square root of a positive number can be a negative number, which is incorrect. The final answer is B.
Explanation
Analyzing the Problem We are given a mathematical reasoning that is claimed to be incorrect, and we need to identify the error. The reasoning is as follows:
$7
sqrt{(x-1)^2 + 2x - 1} = 7
sqrt{x^2 - 2x + 1 + 2x - 1} = 7
sqrt{x^2} = x$
We need to determine why this reasoning is flawed by analyzing each step and comparing it to the given options.
Identifying the Error Let's break down the reasoning step by step:
$7
sqrt{(x-1)^2 + 2x - 1} = 7
sqrt{x^2 - 2x + 1 + 2x - 1} : T hi ss t e p e x p an d s t h es q u a re (x-1)^2 correc tl y a s x^2 - 2x + 1$. 2. $7
sqrt{x^2 - 2x + 1 + 2x - 1} = 7
sqrt{x^2} : T hi ss t e p s im pl i f i es t h ee x p ress i o nin s i d e t h es q u a reroo t b yc an ce l in g t h e -2x an d +2x t er m s , a s w e ll a s t h e +1 an d -1$ terms, which is correct. 3. $7
sqrt{x^2} = x : T hi s i s w h ere t h eerror l i es . T h es q u a reroo t o f x^2 i s t h e ab so l u t e v a l u eo f x , d e n o t e d a s |x| . T h ere f ore ,
sqrt{x^2} = |x| , n o t x$.
Analyzing the Options Now let's analyze the given options:
A. en la última raiz se puede escoger el valor de la incógnita de manera que su cuadrado sea negativo. This option is incorrect because the square of any real number is non-negative.
B. la última igualdad está afirmando que la raíz cuadrada de un número positivo puede ser un número negativo. This option correctly identifies the error. The last equality incorrectly assumes that $
sqrt{x^2} = x , w hi c hi so n l y t r u e f or n o n − n e g a t i v e x . I f x i s n e g a t i v e , t h e n
sqrt{x^2} = |x| = -x$, which is a positive number.
C. dentro de las expresiones que se encuentran en las dos primeras raices cuadradas hay términos negativos. This option is incorrect because the terms inside the square roots are not necessarily negative. For example, if x = 1 , then 0"> ( x − 1 ) 2 + 2 x − 1 = ( 1 − 1 ) 2 + 2 ( 1 ) − 1 = 1 > 0 .
D. en la primera igualdad, el desarrollo del trinomio cuadrado perfecto dentro de la raiz tiene un coeficiente negativo. This option is incorrect because the expansion of the perfect square trinomial is correct. ( x − 1 ) 2 = x 2 − 2 x + 1 , and the coefficient of the x term is indeed negative.
Conclusion The error in the reasoning is that it assumes $
sqrt{x^2} = x , w hi c hi so n l y t r u e f or x \geq 0 . T h ecorrec t s t a t e m e n t i s
sqrt{x^2} = |x|$. Therefore, the correct answer is option B.
Examples
When simplifying expressions involving square roots, it's crucial to remember that x 2 = ∣ x ∣ , not just x . For example, if you're calculating the distance between two points on a number line, say -3 and 2, you would use the absolute value to ensure the distance is positive: d = ∣ − 3 − 2∣ = ∣ − 5∣ = 5 . This concept is also vital in physics when dealing with magnitudes of vectors, ensuring that the magnitude is always a non-negative value. Understanding absolute values ensures accurate calculations and interpretations in various mathematical and real-world contexts.
The reasoning is flawed because it incorrectly assumes that x 2 = x without accounting for the absolute value. The correct formulation is x 2 = ∣ x ∣ . Thus, the correct answer is option B, which identifies that the last equality misstates the nature of square roots.
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