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    I’ve completed the square of x^2+4x+9 as (x+2)^2+5. How do I deduce the maximum value of 1/x^2+4x+9?
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    (Original post by Reece.W.J)
    I’ve completed the square of x^2+4x+9 as (x+2)^2+5. How do I deduce the maximum value of 1/x^2+4x+9?
    I'm assuming you mean 1/(x^2+4x+9). Please use brackets to make it clearer next time.

    To maximise the value of the fraction, you need to minimise the value of the denominator. You know that the denominator is equivalent to (x+2)^2+5. Think about what the minimum possible value of (x+2)^2 is (remember that a negative number squared is positive), then just add 5.
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    (Original post by Reece.W.J)
    I’ve completed the square of x^2+4x+9 as (x+2)^2+5. How do I deduce the maximum value of 1/x^2+4x+9?
    Lets say you had a single variable y that can be any number >0...

    For what value of y is (1/y) a maximum? Not the actual number, but conceptually, how do you make 1/y as big as possible?
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    (Original post by TheMindGarage)
    I'm assuming you mean 1/(x^2+4x+9). Please use brackets to make it clearer next time.

    To maximise the value of the fraction, you need to minimise the value of the denominator. You know that the denominator is equivalent to (x+2)^2+5. Think about what the minimum possible value of (x+2)^2 is (remember that a negative number squared is positive), then just add 5.
    -2?
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    (Original post by Kevin De Bruyne)
    Lets say you had a single variable y that can be any number >0...

    For what value of y is (1/y) a maximum? Not the actual number, but conceptually, how do you make 1/y as big as possible?
    Is it just the 5 from completing the square? Is the fraction just there to try and trick me idk?...
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    (Original post by Reece.W.J)
    -2?
    Nope. If (x+2) squared is -2, there's no real value that (x+2) could have. Try substituting in various values of x (both positive and negative) and see what you get.
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    (Original post by TheMindGarage)
    Nope. If (x+2) squared is -2, there's no real value that (x+2) could have. Try substituting in various values of x (both positive and negative) and see what you get.
    I have no idea tbh I struggle to visualise maths from typing but would it just be the 5 from completing the square? The other question I did the minimum value was -14 and that was the same position as the 5 in this. Does that mean the answer is 5 or is 5 the minimum value
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    (Original post by Reece.W.J)
    I have no idea tbh I struggle to visualise maths from typing but would it just be the 5 from completing the square? The other question I did the minimum value was -14 and that was the same position as the 5 in this. Does that mean the answer is 5 or is 5 the minimum value
    Correct! Basically the minimum value for anything squared (x+2) in this case is zero. For this example, the minimum value is reached when x = -2.

    Now, there's one more step. The original question was the maximum value of 1/(x^2+4x+9), so you need to do 1 divided by 5.

    As a tip for visualising quadratic functions after you complete the square, start with a standard y=x^2 curve. The bit inside the brackets (+2 in this case) is how far the curve is shifted to the left or the right, but be careful - a positive value goes to the left and a negative to the right. The bit outside the brackets (+5) is how far the curve is shifted up or down. In this case, positive means shifted up and negative shifted down. If you still struggle with this, try opening up a graph plotter such as Desmos and playing around with these kinds of graphs.
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    (Original post by TheMindGarage)
    Correct! Basically the minimum value for anything squared (x+2) in this case is zero. For this example, the minimum value is reached when x = -2.

    Now, there's one more step. The original question was the maximum value of 1/(x^2+4x+9), so you need to do 1 divided by 5.

    As a tip for visualising quadratic functions after you complete the square, start with a standard y=x^2 curve. The bit inside the brackets (+2 in this case) is how far the curve is shifted to the left or the right, but be careful - a positive value goes to the left and a negative to the right. The bit outside the brackets (+5) is how far the curve is shifted up or down. In this case, positive means shifted up and negative shifted down. If you still struggle with this, try opening up a graph plotter such as Desmos and playing around with these kinds of graphs.
    Ok so the answer is 1/5 or 0.2 but how come this is the maximum value when the number in the same place on a previous question was the minimum value?
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    (Original post by Reece.W.J)
    Ok so the answer is 1/5 or 0.2 but how come this is the maximum value when the number in the same place on a previous question was the minimum value?
    You have to do 1 divided by the quadratic. 1 divided by a small number gives a large number.
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    (Original post by TheMindGarage)
    You have to do 1 divided by the quadratic. 1 divided by a small number gives a large number.
    Idk
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    (Original post by Reece.W.J)
    Idk
    If you still don't get it, try opening up a graph plotter and plotting the graphs of y=x^2+4x+9 and y=1/(x^2+4x+9) on the same set of axes.
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    (Original post by TheMindGarage)
    If you still don't get it, try opening up a graph plotter and plotting the graphs of y=x^2+4x+9 and y=1/(x^2+4x+9) on the same set of axes.
    Did I do something wrong ?Name:  E702710A-A2DB-4F80-A73F-A20D2D253C38.jpg.jpeg
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    (Original post by Reece.W.J)
    Did I do something wrong ?
    Nope. That's right. You should see that when the top graph is at its minimum point (at -2, 5), the bottom graph is at its maximum point.
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    (Original post by TheMindGarage)
    Nope. That's right. You should see that when the top graph is at its minimum point (at -2, 5), the bottom graph is at its maximum point.
    Yeah so is that 1/5 or 0.2?
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    (Original post by Reece.W.J)
    Yeah so is that 1/5 or 0.2?
    Yep. Hopefully now you see that if you find the minimum point of one graph and do 1 divided by that graph, you get a maximum point.
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    (Original post by TheMindGarage)
    Yep. Hopefully now you see that if you find the minimum point of one graph and do 1 divided by that graph, you get a maximum point.
    I understand now thank you
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