Precalculus : Quadratic Systems
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Example Questions
Example Question #1 : Quadratic Systems
Which of the following could NOT be a possible number of solutions of a system of quadratic and linear equations?
How many times the graphs intersect
3
1
0
2
3
Recall that the solution of a system of equations is given by the intersection points of the graphs. Thus, this question is really asking how many times a parabola and a line can intersect. Visualize a parabola and a line. For this purpose, let's say that the parabola is facing up. If a line is drawn horizontally under the vertex, then it would not intersect the parabola at all, so the system would have no solutions. If the line is tangent to, or just skims the edge, of the parabola then it would only intersect once and the system would have one solution. If the line goes straight through the parabola, then it would intersect twice. There is no other option for the orientation of the line and parabola. Thus, there cannot be three solutions.
Example Question #1 : Conic Sections
Find the coordinate of intersection, if possible: 
To solve for x and y, set both equations equal to each other and solve for x.
Substitute 
The coordinate of intersection is 
Example Question #2 : Conic Sections
Find the intersection(s) of the two parabolas: 
Set both parabolas equal to each other and solve for x.
Substitute both values of 
The coordinates of intersection are:
Example Question #1 : Solve A System Of Quadratic Equations
Find the points of intersection:
To solve, set both equations equal to each other:
To solve as a quadratic, combine like terms by adding/subtracting all three terms from the right side to the left side:
This simplifies to
Solving by factoring or the quadratic formula gives the solutions 
Plugging each into either original equation gives us:
Our coordinate pairs are 
Example Question #2 : Solve A System Of Quadratic Equations
Give the coordinate pairs that satisfy the system of equations.
To solve, set the two quadratics equal to each other and then combine like terms:
Solving by factoring or using the quadratic formula gives us the solutions 
To find the y-coordinates, plug these into either equation:
Example Question #2 : Solve A System Of Quadratic Equations
Give the 
To solve, first re-write the second one so that y is isolated on the left side:
Now set the two quadratics equal to each other:
Using the quadratic formula or by factoring, we get the two solutions 
To get the y-coordinates, plug these numbers into either function:
Example Question #4 : Solve A System Of Quadratic Equations
Find the coordinate pairs satisfying both polynomials:
To solve, set the two polynomials equal to each other:
Solving with the quadratic formula or by factoring gives us the solutions 5 and -3.
To get the y-coordinates, plug these numbers into either of the original equations:
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