Explain the relationship between the discriminant of a quadratic equation, the number and type of solutions?

Question

Explain the relationship between the discriminant of a quadratic equation, the number and type of solutions of the quadratic equation, and the x-intercepts of its corresponding graph?

Answer 1

If the equation is ax^2 + bx +c = 0,
the discriminant is b^2 - 4ac.
This determines the number and type of solutions
as follows:
1). If b^2-4ac>0 there are 2 unequal real roots.
The graph of the equation crosses the x-axis
in 2 distinct places.

2). If b^2-4ac = 0, the equation has 2 equal roots.
The graph is tangent to the x-axis.

3). If b^2-4ac<0 the equation has 2 complex roots
and the graph does not meet the x-axis.

Answer 2

In reality, the nerve also has a role in supplying parts of the heart, windpipe muscles and mucous membranes, and the esophagus, explaining its route. As the recurrent nerve hooks around the subclavian artery or aorta, it gives off several cardiac filaments to the deep part of the cardiac plexus. As it ascends in the neck it gives off branches, more numerous on the left than on the right side, to the mucous membrane and muscular coat of the esophagus; branches to the mucous membrane and muscular fibers of the trachea; and some pharyngeal filaments to the Constrictor pharyngis inferior.” Even apart from this function, there are features that are the result of embryonic development—not because of evolution, but because the embryo develops from a single cell in a certain order. For example, the embryo needs a functioning simple heart early on; this later descends to its position in the chest, dragging the nerve bundle with it.