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Difference between revisions of "1957 AHSME Problems/Problem 14"

(Created page with "If <math>y = \sqrt{x^2 - 2x + 1} + \sqrt{x^2 + 2x + 1}</math>, then <math>y</math> is: <math>\textbf{(A)}\ 2x\qquad \textbf{(B)}\ 2(x+1)\qquad\textbf{(C)}\ 0\qquad\textbf{(D)...")
 
(→‎Solution)
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In order to solve the problem, we will use two properties, namely, that <math>(a+b)^2=a^2+2ab+b^2</math> and <math>(a-b)^2=a^2-2ab+b^2</math>.
 
In order to solve the problem, we will use two properties, namely, that <math>(a+b)^2=a^2+2ab+b^2</math> and <math>(a-b)^2=a^2-2ab+b^2</math>.
  
−
We can use this to simplify the equation, as <math>y = \sqrt{x^2 - 2x + 1} + \sqrt{x^2 + 2x + 1}</math> turns into <math>y = x + 1 + x - 1</math>. <math>y</math> is <math>\boxed{\textbf{(A) }2x}</math>.
+
We can use this to simplify the equation, as <math>y = \sqrt{x^2 - 2x + 1} + \sqrt{x^2 + 2x + 1}</math> turns into <math>y = x + 1 + x - 1</math>. However, the square root function only allows for nonnegative inputs and only generates nonnegative outputs, so <math>y</math> is <math>\boxed{\textbf{(D) }|x+1|+|x-1|}</math>.
 +
 
 
==See Also==
 
==See Also==
 
{{AHSME box|year=1957|num-b=13|num-a=15}}
 
{{AHSME box|year=1957|num-b=13|num-a=15}}
 
{{MAA Notice}}
 
{{MAA Notice}}
 
[[Category:AHSME]][[Category:AHSME Problems]]
 
[[Category:AHSME]][[Category:AHSME Problems]]

Revision as of 22:25, 27 March 2021

If $y = \sqrt{x^2 - 2x + 1} + \sqrt{x^2 + 2x + 1}$, then $y$ is:

$\textbf{(A)}\ 2x\qquad \textbf{(B)}\ 2(x+1)\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ |x-1|+|x+1|\qquad\textbf{(E)}\ \text{none of these}$

Solution

In order to solve the problem, we will use two properties, namely, that $(a+b)^2=a^2+2ab+b^2$ and $(a-b)^2=a^2-2ab+b^2$.

We can use this to simplify the equation, as $y = \sqrt{x^2 - 2x + 1} + \sqrt{x^2 + 2x + 1}$ turns into $y = x + 1 + x - 1$. However, the square root function only allows for nonnegative inputs and only generates nonnegative outputs, so $y$ is $\boxed{\textbf{(D) }|x+1|+|x-1|}$.

See Also

1957 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

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