AMC 10 · 2006 · #8

Grade 8 algebra
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
systems-of-equationsquadratic-equationsvieta-formulas convert-to-algebra ↑ Prerequisites: systems-of-equations
📏 Medium solution 💡 2 insights
Problem
A parabola with equation y=x2+bx+cy = x^2 + bx + c passes through the two points (2,3)(2,3) and (4,3)(4,3). The leading coefficient is fixed at 1, but b and c are unknown constants. Find the value of the constant term c.

Pick an answer.

(A)
2
(B)
5
(C)
7
(D)
10
(E)
11

AMC 10 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

The equation already carries two unknown constants, b and c, so Tool #4 (Introduce a Variable) says: treat them as the quantities to solve for and turn each passing-through condition into an equation. That gives two equations in two unknowns. Tool #7 (Identify Subproblems) breaks the job into stages — first eliminate c to pin down b, then use b to recover c — instead of attacking both at once. Tool #3 (Eliminate Possibilities) is a safety net: c must be one of the five listed values, so any answer outside the choices signals an arithmetic slip.

1STEP 1

Turn each point into an equation

Substituting each point gives 4+2b+c=34 + 2b + c = 3 and 16+4b+c=316 + 4b + c = 3, that is 2b+c=−12b + c = -1 and 4b+c=−134b + c = -13.

4 + 2b + c = 3 → 2b + c = -1; 16 + 4b + c = 3 → 4b + c = -13
2STEP 2

Subtract to cancel c and find b

Both equations hold a lone +c, so subtracting the first from the second wipes it out: 2b=−122b = -12, giving b=−6b = -6.

(4b + c) - (2b + c) = -13 - (-1) → 2b = -12 → b = -6
3STEP 3

Back-substitute to get c

Put b=−6b = -6 back into 2b+c=−12b + c = -1: −12+c=−1-12 + c = -1, so c=11c = 11 — choice (E).

2(-6) + c = -1 → -12 + c = -1 → c = 11 → (E)
Answer
11
Rebuild the parabola with b = -6 and c = 11: y = x² - 6x + 11. Test the two given points — at x = 2: 4 - 12 + 11 = 3, and at x = 4: 16 - 24 + 11 = 3 — both land on y = 3, exactly as required. The constant term c = 11 is the y-intercept, and since the vertex sits at x = 3 with height 9 - 18 + 11 = 2, the curve rises from 2 up to 11 as x moves from 3 out to 0, which is consistent. Value 11 is choice (E).
💡Key takeaway

Plug each point into the equation to get two equations, then subtract them so the shared c cancels — that hands you b, and one more substitution gives c = 11.

  • Turn each point into an equation
  • Subtract to cancel c and find b
  • Back-substitute to get c

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