On a movie set, an archway is modeled by the equation
y = -0.5x2 + 3x, where y is the height in feet and x is the horizontal distance in feet. a laser is directed at the archway at an angle modeled by the equation -0.5x + 2.42y = 7.65 such that the beam crosses the archway at points a and b. at what height from the ground are the points a and b?

Basically we're looking for the intersection of the parabola
y=-0.5x^2+3x,(1)
and the line
-0.5x+2.42y=7.65(2)

To solve for the intersections, we will use the comparison method, expressing equation (2)  in terms of y, then equate the right-hand sides of (1) & (2a).
From (2), solve & isolate y:
y=(7.65+0.5x)/2.42(2a)

Equate y in both (1) & (2a),
-0.5x^2+3x=(7.65+0.5x)/2.42
transposing all terms to the left,
-0.5x^2+2.7934x-3.16116=0
Solve for x using the quadratic equation:
x=(-2.793+/-sqrt(2.793^2-4(-0.5)(-3.16116)))/(2(-0.5))
x1=1.5765 or x2=4.0102
Substituting the two values of x (x1 & x2) into (1) will give the corresponding height (y) of the laser beam.

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