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Solving Quadratic Systems

Solving Quadratic Systems 1.1 - Solution

arrow_back Return to Solving Quadratic Systems

To solve the system of equations by graphing, we will draw the graph of the two quadratic functions on the same coordinate grid. Let's start with the first equation.

Graphing the First Parabola

To graph the parabola we first need to identify and For this equation we have that and Now, we can find the vertex using its formula. To do this, we will need to think of as a function of Let's find the -coordinate of the vertex.
We use the -coordinate of the vertex to find its -coordinate by substituting it into the given equation.
Simplify right-hand side
The -coordinate of the vertex is Thus, the vertex is at the point With this, we also know that the axis of symmetry of the parabola is the line Next, let's find two more points on the curve, one on each side of the axis of symmetry.

Both and are on the graph. Let's form the parabola by connecting these points and the vertex with a smooth curve.

Graphing the Second Parabola

Just like above, we will start with identifying and For this equation we have that and Let's again find the -coordinate of the vertex.
We use the -coordinate of the vertex to find its -coordinate by substituting it into the given equation.
Simplify right-hand side
The -coordinate of the vertex is Thus, the vertex is at the same point as the previous parabola. The next step is to find two more points on the curve, one on each side of the axis of symmetry, which is

Both and are on the graph. Let's form the second parabola by connecting these points and the vertex with a smooth curve.

Finding the Solutions

Finally, let's try to identify the coordinates of the points of intersection of the two parabolas.

It looks like the point of intersection occurs at

Checking the Answer

To check our answer, we will substitute the values of the point of intersection in both equations of the system. If they produce true statements, our solution is correct.
,
Simplify right-hand side
Calculate power
Multiply
Add and subtract terms
Equation (I) and Equation (II) both produced true statements. Therefore, is the correct solution.