

When you are working with quadratics, you are primarily working
with
ax^{2} + bx + c = 0
or y = ax^{2} + bx + c (where a, b and c are constants). 
A linear quadratic system is a system containing one linear equation and one quadratic equation
(which is generally one straight line and one parabola).
A simple linear system contains two linear equations (which is two straight lines).
When dealing with a straight line and a parabola, there are three possible ways they
may appear on a graph, giving three possible solution situations.
Possible Solution Situations
LinearQuadratic System (line and parabola)
A solution is a location where the straight line and the parabola intersect (cross). 
Situation 1:
When graphed, most linear quadratic systems
will show the line and the parabola intersecting in two points, as seen at the right.
Two solutions


Situation 2:
If the straight line is tangent to the parabola, it will intersect (hit) the parabola in only one location, as seen at the right.
One solution


Situation 3:
It is possible that the straight line and the parabola never touch one another. They do not intersect.
No solutions




It is also possible when working with quadratics, that you may encounter a quadratic where both the x and y variables are squared (with the same coefficients); a circle. 
When dealing with a straight line and a circle, there are also three possible ways they
may appear on a graph, giving three possible solution situations.
Possible Solution Situations
LinearQuadratic System (line and circle)
A solution is a location where the straight line and the circle intersect (cross). 
Situation 1:
When graphed, most linear quadratic systems
will show the line and the circle intersecting in two points, as seen at the right.
Two solutions


Situation 2:
If the straight line is tangent to the circle, it will intersect (hit) the circle in only one location, as seen at the right.
One solution


Situation 3:
It is possible that the straight line and the circle never touch one another. They do not intersect.
No solutions


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