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What Do Linear Quadratic Systems Look Like?
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statement When you are working with quadratics, you are primarily working with
ax
2 + bx + c = 0  or  y = ax2 + 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.

bullet Possible Solution Situations
Linear-Quadratic 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

graph3
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

 

graph4
Situation 3:
It is possible that the straight line and the parabola never touch one another. They do not intersect.

No solutions

graph5




statement 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.

bullet Possible Solution Situations
Linear-Quadratic 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


graph3
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

 

graph4
Situation 3:
It is possible that the straight line and the circle never touch one another. They do not intersect.

No solutions

graph5



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