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2006 AMC 10A Problem 8

Problem 8 of 25EasierAlgebraGeometry

A parabola with equation y=x2+bx+cy = x^2 + bx + c passes through the points (2,3)(2, 3) and (4,3).(4, 3). What is c?c?

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Solution

Substituting the points gives 3=4+2b+c3 = 4 + 2b + c and 3=16+4b+c.3 = 16 + 4b + c. Subtracting yields 0=12+2b,0 = 12 + 2b, so b=−6.b = -6. Then c=3−4−2(−6)=11.c = 3 - 4 - 2(-6) = 11. Thus, the correct answer is E.

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Concepts: parabola · system of equations

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.