Difference between revisions of "2006 AMC 10A Problems/Problem 8"
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== Problem == | == Problem == | ||
| − | A parabola with equation <math>\displaystyle y=x^2+bx+c</math> passes through the points (2,3) and (4,3). What is <math>\displaystyle c</math>? | + | A [[parabola]] with equation <math>\displaystyle y=x^2+bx+c</math> passes through the points (2,3) and (4,3). What is <math>\displaystyle c</math>? |
<math> \mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 5\qquad \mathrm{(C) \ } 7\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 11 </math> | <math> \mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 5\qquad \mathrm{(C) \ } 7\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 11 </math> | ||
== Solution == | == Solution == | ||
| − | Substitute the points (2,3) and (4,3) into the | + | Substitute the points (2,3) and (4,3) into the given equation for (x,y). |
Then we get a system of two equations: | Then we get a system of two equations: | ||
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<math>0=1+-12+c</math> | <math>0=1+-12+c</math> | ||
| − | <math>c=11</math>. E is the answer. | + | <math>c=11</math>. So <math>\mathrm{(E) \ }</math> is the answer. |
== See Also == | == See Also == | ||
*[[2006 AMC 10A Problems]] | *[[2006 AMC 10A Problems]] | ||
Revision as of 10:26, 31 July 2006
Problem
A parabola with equation 
passes through the points (2,3) and (4,3). What is 
?
Solution
Substitute the points (2,3) and (4,3) into the given equation for (x,y).
Then we get a system of two equations:
Subtracting the first equation from the second we have:
Then using in the first equation:
. So 
is the answer.
