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Quadratic equation - how to understand the impact of the middle term?

  1. Oct 30, 2013 #1

    I am trying understand if there is a quick way to figure out the impact of the X term in a quadratic equation.

    For example, by looking at the following equation, I know that it is a parabola (X^2) ; by looking at 24, I know that it is the y-intercept. However, I don't know what is the impact of the middle term (10X). I don't know what will happen to the curve if I change it to 20X. I know that the parabola will shift to the left or right but not sure exactly.

    y = X^2 + 10X +24

    Is there a way to figure of the effect of the middle term?

  2. jcsd
  3. Oct 30, 2013 #2


    Staff: Mentor

    In your equation, the 10x term causes a translation to the left by 5 units.

    Your equation can be rewritten as y = x2 + 10x + 25 - 1 = (x + 5)2 - 1.

    Relative to the graph of y = x2, the equation above is shifted left by 5 units and down 1 unit. Instead of the vertex being at (0, 0), the vertex is now at (-5, -1). The point (1, 1) on the graph of y = x2 has also been shifted left and down, and is now at (-4, 0).
  4. Oct 31, 2013 #3


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    I'm assuming you are looking for only the contribution the middle term brings.

    Instead of considering 10x, it's easier to consider half of the middle term: that is five times 2x. Then
    ##x^2 + 2 a x + 24 = (x+a)^2 + 24 - a^2##.
    So it takes the polynomial ##x^2 + 24## and translates it left a units and down a2 units. In your example, a=5.
  5. Oct 31, 2013 #4


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    Dearly Missed

    Note therefore, which ought to be clear from the two previous posts, that the middle term has no effect on the SHAPE of the parabola, merely on its positioning in the plane.
  6. Nov 5, 2013 #5
    I am not sure what you mean, but your question was:
    y = X^2 + 10X +24
    well i use factorization because it can be factorized:
    so y = X^2 + 6X+4X +24
    y= X(X+6) +4(X+6)
    y= (X+4) (X+6)
    EITHER X IS = -4 OR X IS = -6
  7. Nov 5, 2013 #6


    Staff: Mentor

    The OP wasn't asking for the x-intercepts, which you have found. He/she was asking about what role the 10x term plays in the shape of the graph of this parabola.
  8. Mar 13, 2014 #7
    the middle term has no effect on shape of parabola,
    while graphing x^2+10x+24=0
    y=x^2 and y=-10x-24
    middle term plays a role in where the linear equation cuts the parabola and not on the shape of the parabola
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