# Find the Points of Intersection of a Parabola with a Line

This is tutorial on finding the points of intersection of a parabola with a line; general solution.

## Example 1

Find the points of intersection of the parabola with the line given respectively by their equations

y = 2 x 2 + 4 x - 3
2y + x = 4
Solution to Example 1

We first solve the linear equation for y as follows:
y = - (1 / 2) x + 2
We now substitute y in the equation of the parabola by - (1 / 2) x + 2 as follows
- (1 / 2) x + 2 = 2 x
2 + 4 x - 3
We now group like terms
2 x
2 + (9 / 2) x - 5 = 0
Solve the above quadratic equation for x to obtain two solutions
x = (- 9 - √(241)) / 8 and x = (- 9 + √(241)) / 8
We now substitute the values of x obtained above into the equation y = - (1 / 2) x + 2 to obtain the values for y as follows
y = (41 + √(241)) / 16
and y = (41 - √(241)) / 16
The two points of intersection of the two circless are given by
((- 9 - √(241)) / 8 , (41 + √(241)) / 16 ) and ((- 9 + √(241)) / 8 , (41 - √(241)) / 16 )
Approximated as:(-3.06 , 3.53 ) and (0.82 , 1.59)

Shown below is the graph of the parabola, the line and the two points of intersection. ## More References and Links to Parabola

Find Points Of Intersection of Parabola and Line - Calculator.
Interactive tutorial on the
Equation of a Parabola.
Interactive tutorial on how to
find the equation of a parabola.
Three Points Parabola Calculator.