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Thread: Minimum point on a quadratic curve

  1. #1
    Senior Member Mukilab's Avatar
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    Minimum point on a quadratic curve

    How do I find the minimum point on a quadratic curve other than using the completing the square method e.g. I'm presented with x^2+10x+21 and it cannot be done using the above method.

    Please help me, otherwise how would I find a maximum point (by any method lower than A-Level).

    Thansk.
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  2. #2
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    e^(i*pi)'s Avatar
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    Quote Originally Posted by Mukilab View Post
    How do I find the minimum point on a quadratic curve other than using the completing the square method e.g. I'm presented with x^2+10x+21 and it cannot be done using the above method.

    Please help me, otherwise how would I find a maximum point (by any method lower than A-Level).

    Thansk.
    You can complete the square on any quadratic - even the quadratic formula is derived by completing the square. I suspect they want you to complete the square because the only other way is to find $\displaystyle \dfrac{dy}{dx} = 0$ and then check the sign of $\displaystyle \dfrac{d^2y}{dx^2}$. In other words calculus which is not lower than A level.

    Completed the square form is $\displaystyle (x+k)^2 + h$. The lowest point on the graph is equal to $\displaystyle h$ because $\displaystyle (x+k)^2 \geq 0$

    $\displaystyle ax^2+bx+c$ into completed square form is $\displaystyle \left(x+\dfrac{b}{2a}\right)^2 - \dfrac{b^2}{4a^2} + c$.

    Aligning that with the $\displaystyle (x+k)^2 + h$ form we get $\displaystyle k = \dfrac{b}{2a}$ and $\displaystyle h = \dfrac{b^2}{4a^2} + c$

    In your case you have $\displaystyle a=1$ , $\displaystyle b = 10$ and $\displaystyle c=21$


    My answer is in the spoiler but if you have any issues getting there please post your workings

    Spoiler:
    $\displaystyle (x+5)^2-4$ - this means -4 is the minimum point and this happens when $\displaystyle x = -5$ so the minimum point is $\displaystyle (-5,-4)$
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  3. #3
    Senior Member Mukilab's Avatar
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    My thanks, that formula of k= and h= is very clear and concise and will serve me throughout the future year I suspect

    I guess I still have to learn the A-Level method soon

    Thank you!
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