What Values of K Make the Intersection of Two Graphs Positive?

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The discussion centers on finding the values of k for which the line y = kx + 3 intersects the parabola y = x^2 + 8x at two distinct points. Participants emphasize the importance of setting the equations equal to each other and using the quadratic formula to analyze the discriminant, which determines the nature of the roots. The discriminant is expressed as (k - 8)^2 + 12, and it is established that this expression is always positive for any value of k. Therefore, the conclusion is that the line will always intersect the parabola at two distinct points regardless of the value of k. This leads to the understanding that the intersection condition is satisfied for all k.
  • #31
Try completing the square: x^2- 16k+ 76= (x- ?)^2+ ?. Now, what values of k make that positive?
 
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  • #32
HallsofIvy said:
Try completing the square: x^2- 16k+ 76= (x- ?)^2+ ?. Now, what values of k make that positive?
Been there and done that. See post #27.
 

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