# 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.