Difference between revisions of "1958 AHSME Problems/Problem 23"
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== Problem == | == Problem == | ||
| − | If, in the expression <math> x^2 | + | If, in the expression <math> x^2 - 3</math>, <math> x</math> increases or decreases by a positive amount of <math> a</math>, the expression changes by an amount: |
| − | <math> \textbf{(A)}\ {\pm 2ax | + | <math> \textbf{(A)}\ {\pm 2ax + a^2}\qquad |
\textbf{(B)}\ {2ax \pm a^2}\qquad | \textbf{(B)}\ {2ax \pm a^2}\qquad | ||
| − | \textbf{(C)}\ {\pm a^2 | + | \textbf{(C)}\ {\pm a^2 - 3} \qquad |
| − | \textbf{(D)}\ {(x | + | \textbf{(D)}\ {(x + a)^2 - 3}\qquad\\ |
| − | \textbf{(E)}\ {(x | + | \textbf{(E)}\ {(x - a)^2 - 3}</math> |
| + | == Solution == | ||
| + | Let us represent the increase or decrease in <math>x</math> by <math>(x \pm a)</math> | ||
| − | + | Thus our original expression becomes | |
| − | <math>\fbox{}</math> | + | <cmath>(x \pm a)^2 - 3</cmath> |
| + | <cmath>x^2 \pm 2ax + a^2 - 3</cmath> | ||
| + | The absolute difference between these two expressions is <math>\pm 2ax + a^2</math>. | ||
| + | |||
| + | Therefore, the answer is <math>\fbox{(A)}</math> | ||
== See Also == | == See Also == | ||
Latest revision as of 01:00, 22 December 2015
Problem
If, in the expression 
, 
increases or decreases by a positive amount of 
, the expression changes by an amount:
Solution
Let us represent the increase or decrease in by 
Thus our original expression becomes
![\[(x \pm a)^2 - 3\]](http://latex.artofproblemsolving.com/a/6/b/a6b9a46c3a53cdb7f52ebdc307c2c866c4141e32.png)
![\[x^2 \pm 2ax + a^2 - 3\]](http://latex.artofproblemsolving.com/8/2/c/82c5bae5af01d973db66f5e9d1e41511c5d1287a.png)
The absolute difference between these two expressions is 
.
Therefore, the answer is 
See Also
| 1958 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 22 |
Followed by Problem 24 | |
| 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. 
