Difference between revisions of "1986 AHSME Problems/Problem 13"
(Created page with "==Problem== A parabola <math>y = ax^{2} + bx + c</math> has vertex <math>(4,2)</math>. If <math>(2,0)</math> is on the parabola, then <math>abc</math> equals <math>\textbf{(A)}...") |
(Added a solution with explanation) |
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==Solution== | ==Solution== | ||
| − | + | Consider the quadratic in completed square form: it must be <math>y=a(x-4)^{2}+2</math>. Now substitute <math>x=2</math> and <math>y=0</math> to give <math>a=-\frac{1}{2}</math>. Now expanding gives <math>y=-\frac{1}{2}x^{2}+4x-6</math>, so the product is <math>-\frac{1}{2} \cdot 4 \cdot -6 = 3 \cdot 4 = 12</math>, which is <math>\boxed{E}</math>. | |
== See also == | == See also == | ||
Latest revision as of 17:35, 1 April 2018
Problem
A parabola 
has vertex 
. If 
is on the parabola, then 
equals
Solution
Consider the quadratic in completed square form: it must be 
. Now substitute 
and 
to give 
. Now expanding gives 
, so the product is 
, which is 
.
See also
| 1986 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 12 |
Followed by Problem 14 | |
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
