Asked by nb
How many solutions exist for the system of equations in the graph?
On a coordinate plane, a graph of a circle and a parabola are shown. The circle has a center at (0, 0) and a radius of 8 units. The parabola opens up and goes through (0, 8), has a vertex at (3.5, 2), and goes through (7, 8).
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GPT-5 mini
AI
Two.
Work (brief): The parabola is y = a(x − 3.5)^2 + 2 and using (0,8) gives a = 24/49, so y = (24/49)(x − 3.5)^2 + 2. Substitute into x^2 + y^2 = 64 to get a quartic in t = x − 3.5. That quartic has the root t = −3.5 (x = 0, the point (0,8)), and dividing it by (t + 3.5) leaves the cubic
2304 t^3 − 8064 t^2 + 56644 t − 131026 = 0.
Its derivative 6912 t^2 − 16128 t + 56644 has negative discriminant, so the cubic is monotone and has exactly one real root. Therefore the quartic has exactly two real roots (one is t = −3.5), so the circle and parabola meet in two points.
Work (brief): The parabola is y = a(x − 3.5)^2 + 2 and using (0,8) gives a = 24/49, so y = (24/49)(x − 3.5)^2 + 2. Substitute into x^2 + y^2 = 64 to get a quartic in t = x − 3.5. That quartic has the root t = −3.5 (x = 0, the point (0,8)), and dividing it by (t + 3.5) leaves the cubic
2304 t^3 − 8064 t^2 + 56644 t − 131026 = 0.
Its derivative 6912 t^2 − 16128 t + 56644 has negative discriminant, so the cubic is monotone and has exactly one real root. Therefore the quartic has exactly two real roots (one is t = −3.5), so the circle and parabola meet in two points.
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