mathleaks.com mathleaks.com Start chapters home Start History history History expand_more
{{ item.displayTitle }}
navigate_next
No history yet!
Progress & Statistics equalizer Progress expand_more
Student
navigate_next
Teacher
navigate_next
Expand menu menu_open Minimize
{{ filterOption.label }}
{{ item.displayTitle }}
{{ item.subject.displayTitle }}
arrow_forward
No results
{{ searchError }}
search
menu_open home
{{ courseTrack.displayTitle }}
{{ statistics.percent }}% Sign in to view progress
{{ printedBook.courseTrack.name }} {{ printedBook.name }}
search Use offline Tools apps
Login account_circle menu_open

Solving Linear-Quadratic Systems

Solving Linear-Quadratic Systems 1.3 - Solution

arrow_back Return to Solving Linear-Quadratic Systems

To solve the system of equations by graphing, we will draw the graph of the quadratic function and the linear function on the same coordinate grid. Let's start with the parabola.

Graphing the 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 will 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 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 Line

Let's now graph the linear function on the same coordinate plane. For a linear equation written in slope-intercept form, we can identify its slope and -intercept The slope of the line is and the intercept is

Finding the Solutions

Finally, let's try to identify the coordinates of the points of intersection of the parabola and the line.

It looks like the only point of intersection occurs at

Checking Our Answer

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