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.
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