Difference between revisions of "2006 AMC 10A Problems/Problem 8"
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A [[parabola]] with equation <math>y=x^2+bx+c</math> passes through the points <math> (2,3) </math> and <math> (4,3) </math>. What is <math>c</math>? | A [[parabola]] with equation <math>y=x^2+bx+c</math> passes through the points <math> (2,3) </math> and <math> (4,3) </math>. What is <math>c</math>? | ||
| − | <math> \ | + | <math> \textbf{(A) } 2\qquad \textbf{(B) } 5\qquad \textbf{(C) } 7\qquad \textbf{(D) } 10\qquad \textbf{(E) } 11 </math> |
== Solution == | == Solution == | ||
Revision as of 22:00, 16 December 2021
Contents
Problem
A parabola with equation 
passes through the points 
and 
. What is 
?
Solution
Solution 1
Substitute the points and into the given equation for 
.
Then we get a system of two equations:
Subtracting the first equation from the second we have:
Then using in the first equation:
is the answer.
Solution 2
Alternatively, notice that since the equation is that of a conic parabola, the vertex is likely 
. Thus, the form of the equation of the parabola is 
. Expanding this out, we find that 
.
Solution 3
The points given have the same 
-value, so the vertex lies on the line 
.
The 
-coordinate of the vertex is also equal to 
, so set this equal to 
and solve for 
, given that 
:
Now the equation is of the form 
. Now plug in the point and solve for :
Solution 4
Substituting y into the two equations, we get:
Which can be written as:
4, 2, are the solutions to the quadratic. Thus:
See also
| 2006 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 7 |
Followed by Problem 9 | |
| 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 | ||
| All AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
