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Learning intentions:
In this section we will examine:
- Solving simultaneous equations involving one quadratic and one linear equation.
- How many solutions exist for the simultaneous equations using the discriminate.
Simultaneous quadratic equations
Solving simultaneous equations involves finding a set of values (coordinates) that satisfy both equations. Graphically, this set of values is the point of intersection when the two equations are graphed. In this section we will look at finding the points of intersection between a quadratic and a linear equation by solving the system simultaneously. Depending on the equations of the quadratic and linear equations, it is possible to have:
- No points of intersection.
- One point of intersection (the linear equation is a tangent to the parabola).
- Two points of intersection.
The discriminant can be used to determine how many points of intersection exist.
Figure 1 - Graphs of simultaneous equations have 0, 1 or 2 solutions.
To solve the system of simultaneous equations you must have both in the form y = Rule.
4H - VIDEO EXAMPLE 1:
Find the point(s) of intersection between the parabola y = x2 + 6x - 3 and the line y = x + 3.
4H - VIDEO EXAMPLE 2:
Use the discriminant to find when the intersection between y = 2x2 + 2 and y = mx will have no solutions, one solution and two solutions.
- Adjust the values of m in the GeoGebra worksheet below to investigate for what values of m the curves y = 2x2 + 2 and y = mx have 0, 1 and 2 points of intersection.
Success criteria:
You will be successful if you can:
- Find and state the coordinates of the point(s) of intersection between a linear and quadratic function.
- Determine for what values of a parameter a simultaneous equation will have 0, 1 or 2 solutions.
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