Question

the line with equation x-3y-27=0 meets the parabola y^2=4x at two points. find their coordinates

Answer 1

well, we know that 4x=y^2. This means that x=y^2/4. We can plug that into the first equation and get (y^2/4)-3*y-27=0. This is equivalent to y^2-12y-108=0. Using the quadratic equation, we can derive the following: (y-18)*(y+6)=0. This means that the two roots of y are 18 and -6. Plugging this back into the second equation, we get the two values of x as 81 and 9, respectively. We can plug these back into the first equation just to check and we get 0. Therefore the solutions are (x,y)=(81,18) and (9,-6). Does this make sense?