Difference between revisions of "1964 AHSME Problems/Problem 24"
Talkinaway (talk | contribs) (Created page with "== Problem == Let <math>y=(x-a)^2+(x-b)^2, a, b</math> constants. For what value of <math>x</math> is <math>y</math> a minimum? <math>\textbf{(A)}\ \frac{a+b}{2} \qquad \tex...") |
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Expanding the quadratic and collecting terms gives <math>y = 2x^2 - (2a + 2b)x + (a^2+b^2)</math>. For a quadratic of the form <math>y = Ax^2 + Bx + C</math> with <math>A>0</math>, <math>y</math> is minimized when <math>x = -\frac{B}{2A}</math>, which is the average of the roots. | Expanding the quadratic and collecting terms gives <math>y = 2x^2 - (2a + 2b)x + (a^2+b^2)</math>. For a quadratic of the form <math>y = Ax^2 + Bx + C</math> with <math>A>0</math>, <math>y</math> is minimized when <math>x = -\frac{B}{2A}</math>, which is the average of the roots. | ||
− | Thus, the quadratic is minimized when <math>x = \frac{2a+2b}{2} = \frac{a+b}{2}</math>, which is answer <math>\boxed{\textbf{(A)}}</math>. | + | Thus, the quadratic is minimized when <math>x = \frac{2a+2b}{2\cdot 2} = \frac{a+b}{2}</math>, which is answer <math>\boxed{\textbf{(A)}}</math>. |
==Solution 2== | ==Solution 2== |
Revision as of 02:18, 24 July 2019
Contents
Problem
Let constants. For what value of is a minimum?
Solution 1
Expanding the quadratic and collecting terms gives . For a quadratic of the form with , is minimized when , which is the average of the roots.
Thus, the quadratic is minimized when , which is answer .
Solution 2
The problem should return real values for and , which eliminates and . We want to distinguish between options , and testing should do that, as answers will turn into , respectively.
PLugging in gives , or . This has a minimum at , or at . This is answer .
See Also
1964 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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 | ||
All AHSME Problems and Solutions |
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