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Math Help - Find Vertical Line Intersecting Parabola Only Once

  1. #1
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    Find Vertical Line Intersecting Parabola Only Once

    I have a question with a solution to it but I can't seem to understand exactly how the solution is obtained. This question is related to Geometry.

    The question is: Find all non-vertical lines that pass through the point Q(0,4) and have exactly one point in common with the Parabola y=x^2 + 7x + 20

    Once the equation y=mx+4 is equated to y=x^2 +7x +20, the solution
    x^2 + x(7-m) +16 = 0 is obtained.

    Then it says: "This equation has exactly one solution when (7-m)^2 -4(16) = 0 and this gives: m-7=+-8, m=-1 or m=15

    How do you get from x^2 + x(7-m) +16 = 0
    to (7-m)^2 -4(16) = 0? Thank you.
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  2. #2
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    re: Find Vertical Line Intersecting Parabola Only Once

    Quote Originally Posted by virussss123 View Post
    I have a question with a solution to it but I can't seem to understand exactly how the solution is obtained. This question is related to Geometry.

    The question is: Find all non-vertical lines that pass through the point Q(0,4) and have exactly one point in common with the Parabola y=x^2 + 7x + 20

    Once the equation y=mx+4 is equated to y=x^2 +7x +20, the solution
    x^2 + x(7-m) +16 = 0 is obtained.

    Then it says: "This equation has exactly one solution when (7-m)^2 -4(16) = 0 and this gives: m-7=+-8, m=-1 or m=15

    How do you get from x^2 + x(7-m) +16 = 0
    to (7-m)^2 -4(16) = 0? Thank you.
    For any quadratic equation \displaystyle ax^2 + bx + c = 0 there is exactly one solution when the discriminant \displaystyle b^2 - 4ac = 0.
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  3. #3
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    re: Find Vertical Line Intersecting Parabola Only Once

    Quote Originally Posted by Prove It View Post
    For any quadratic equation \displaystyle ax^2 + bx + c = 0 there is exactly one solution when the discriminant \displaystyle b^2 - 4ac = 0.
    Thank you very much!!! I didn't realise that they worked out the discriminant. Once again, thanks!
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