Difference between revisions of "1957 AHSME Problems/Problem 14"
Angrybird029 (talk | contribs) (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)...") |
m (typo fix) |
||
(One intermediate revision by one other user not shown) | |||
Line 5: | Line 5: | ||
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{( | + | 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 50p box|year=1957|num-b=13|num-a=15}} |
{{MAA Notice}} | {{MAA Notice}} | ||
[[Category:AHSME]][[Category:AHSME Problems]] | [[Category:AHSME]][[Category:AHSME Problems]] |
Latest revision as of 08:21, 25 July 2024
If , then is:
Solution
In order to solve the problem, we will use two properties, namely, that and .
We can use this to simplify the equation, as turns into . However, the square root function only allows for nonnegative inputs and only generates nonnegative outputs, so is .
See Also
1957 AHSC (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 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.