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February 9th, 2018, 10:43 PM   #1
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Exclamation Quadratics


Find the coordinate of point A in the figure. Can you solve it in a way that doesn't involve derivation?
Please solve step by step.

Last edited by skipjack; February 10th, 2018 at 02:46 AM.
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February 9th, 2018, 10:55 PM   #2
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Can you upload the image?
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February 10th, 2018, 02:59 AM   #3
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You were mistyping the image address. If you pasted it, check what was pasted. I've corrected it for you.

As the blue line through the points (-2, 0) and (0,-2) has equation y = -x - 2,
it intersects the parabola at the point (3, -5).

Hence the dotted line through A has equation y = x - 8,
which implies that A is the point (8,0).
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February 10th, 2018, 10:22 AM   #4
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The parabola intersects the x-Axis at:
f(x)=(x^2)+4
0=(x^2)+4 /-4
4=(x^2) /√
±2=x

Since we are looking for the left zero point, it has to be deduced that the left zero point has the coordinate (-2|0).

Now, in order to deduce how the line's function looks like, we'll use the general function for lines:
g(x)=m*x+n
Since the line intersects the y-Axis at y=(-2), the new function looks like that:
g(x)=m*x-2

Since we know the points P(-2|0) and Q(0|-2), the slope will be, |-2|/|-2|=1 and because the tendency is negative, the slope is -1.
Hence the function of the line g is g(x)=-x-2
Thus the line g intersects the parabola at (3|-5)

We are now looking for the function of the perpendicular line h to line g through (3|-5).
Two lines are perpendicular if the slope of the first one equals the negative inverse of the second slope: m1=-(1/m2)
Thus the line h's slope will be 1{=-[1/(-1)]}
Having the slope, we can deduce that the line h will intersect the y-Axis at (0|-8) (-5-3=-8).

Finally, having the line h's function (h(x)=x-8) we can calculate the zero point of the function:
0=x-8 /+8
8=x
Hence, the zero point stays at Z(8|0), which is the sought point.

Hope you like the explanation.

Last edited by skipjack; February 10th, 2018 at 03:11 PM. Reason: to disable smilies
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February 10th, 2018, 10:24 AM   #5
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All the smilies stay for -8
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