Question

complex numbers & higher order polynomials
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how many solutions exist for the quadratic polynomial in the graph?
(1 point)
□ real solutions and □ non - real solutions
check answer remaining attempts 3

Explanation:

Step1: Recall solution - graph relation

The solutions of a quadratic polynomial (roots) are the x - intercepts of its graph (where \(y = 0\)). For a quadratic function \(y=ax^{2}+bx + c\), the number of real solutions is determined by how many times the graph intersects the x - axis. Non - real solutions occur when the graph does not intersect the x - axis (and we use the discriminant \(D=b^{2}-4ac\), but from the graph, we can directly observe intersections).

Step2: Analyze the given graph

Looking at the provided graph of the quadratic polynomial, we can see that the parabola does not intersect the x - axis (the x - axis is the line \(y = 0\), and the graph of the parabola is entirely below or above? Wait, in the graph, the parabola opens downward? Wait, no, the arrow on the y - axis is pointing down, so the parabola has a maximum at \(x = 0\) and the graph is below the x - axis (since the vertex is at \((0,-1)\) or so? Wait, the x - axis is the horizontal line through \(y = 0\). The graph of the quadratic does not cross the x - axis. So, the number of real solutions (x - intercepts) is 0. Since a quadratic polynomial has a degree of 2, by the fundamental theorem of algebra, the total number of solutions (real and non - real, counting multiplicities) is 2. So if there are 0 real solutions, the number of non - real solutions is \(2-0 = 2\). Wait, no, wait the graph: let's re - examine. Wait, the x - axis is the horizontal line with \(y = 0\). The parabola in the graph: does it touch or cross the x - axis? Looking at the graph, the parabola is below the x - axis (the vertex is at \((0, - 1)\) approximately, and the arms go down? Wait, no, the y - axis arrow is pointing down, so the parabola opens downward? Wait, no, if the vertex is the highest point, and the graph is below the x - axis, that means it never crosses the x - axis. So real solutions (x - intercepts) are 0, non - real solutions are 2. Wait, but maybe I made a mistake. Wait, the problem is asking "How many solutions exist for the quadratic polynomial in the graph? real solutions and non - real solutions". Wait, the graph: let's check the x - axis (where \(y = 0\)). The parabola does not intersect the x - axis, so the number of real solutions (x - intercepts) is 0. Since a quadratic has degree 2, the number of solutions (in complex numbers) is 2. So non - real solutions are 2.

Answer:

0 real solutions and 2 non - real solutions