Solving Linear-Quadratic Systems — Quiz

Question 4 • Mathematics II

Which graph most likely shows a system of equations with no solutions?

Answer

Image with description On a coordinate plane, the scale on the x and the y axes ranges from negative 5 to 5 in increments of 1. A parabola opens up and has the vertex at 0 comma 1. A straight line has a positive slope, passing through the points negative 2 comma 0 and 0 comma 2.

Image with description On a coordinate plane, the scale on the x and the y axes ranges from negative 5 to 5 in increments of 1. A parabola opens up and has the vertex at 0 comma negative 1. A straight line has a negative slope, passes through the points 1 comma negative 1 and 0 comma 1.

Image with description On a coordinate plane, the scale on the x and the y axes ranges from negative 5 to 5 in increments of 1. A parabola opens up and has the vertex at 0 comma negative 3. A straight line has a positive slope, passing through the points 0 comma negative 3 and 2 comma negative 2.

Image with description On a coordinate plane, the scale on the x and the y axes ranges from negative 5 to 5 in increments of 1. A parabola opens up and has the vertex at 0 comma negative 1. A straight line has a positive slope, passing through the points 0 comma negative 3 and 3 comma 0.

Explanation

A system has no solutions when the graphs do not intersect. In choice D, the line through (0, −3) and (3, 0) is y = x − 3, while the parabola with vertex (0, −1) is y = x² − 1. Setting them equal gives x² − 1 = x − 3, or x² − x + 2 = 0; its discriminant is 1 − 8 = −7, so there are no real intersections. Thus D shows a system with no solutions.

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