Math Is Fun Forum

ACP45

Member

Registered: 2008-01-14

Posts: 1

Feeling old, kid stumped me. x+y=-12=[xy=24] help!

My 8th grader needed help, but there may just be to much gray on the head, and to little, in it.
if x+y=-12 can the same xy=24?

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Feeling old, kid stumped me. x+y=-12=[xy=24] help!

Yes, one way is to have one number between -2 and -3, and the other number between -10 and -9.
See how (-2)(-10) is less than 24 by 4 and (-3)(-9) is bigger than 24 by 3.
Should be an answer in between.

Your xy=k equation is a hyperbola bound by vertical and horizontal axes.
The x + y = s equation is a diagonal line.

(off_the_subject: If both these graphs were turned 45 degrees or 1/8th of a turn, then the
straight line would be vertical or horizontal and the hyperbola would be
somehow converted to the y^2/t - x^2/t = v type of equation, another kind of hyperbola)

xy = 24

y = 24/x

x + y = -12

x + 24/x = -12

x ( x + 24/x) = -12x

x^2 + 24 = -12x

x^2 + 12x + 24 = 0

At this point I thought I made a mistake.
But probably not, just looks like it.
Try to factor it before resorting to quadratic equation.
See how it is +12x, and not -12x, that's why I thought it was wrong since x + y = -12,
but then I realized that the factors are approximately (x + 2.5)(x + 9.5) = 0, which
then makes x go NEGATIVE, like our first guess.

Do you know the quadratic formula for this type of zero_equals_equation like ax^2 + bx + c = 0  ???

(a=1)x^2 + (b=12)x + (c=24) = 0

quadratic equation is:

x = -b +/- squareRootOf(b^2 - 4ac), all divided by 2a.     

That's 2 answers possible, sometimes one makes sense and the other one is the
square root of a negative number, so you usually ignore that one unless you
have learned about complex numbers with "i" in it.

Plug a,b,c in the formula:
x = -12 +/- squareRootOf(12^2 - 4(1)(24)), all divided by 2(1)

x = -12 +/- squareRootOf(144 - 96), all divided by 2

x = -12 +/- squareRootOf(48), all divided by 2

By the way 48 = 16 times 3 and 16 is a perfect square = 4 times 4.

x = -12 +/- (4)(squareRootOf(3)), all divided by 2

Bring the divide by 2 into both terms since it works out well.

x = -12/2 +/- (4/2)(squareRootOf(3))

x = -6 +/- 2 SquareRootsOf(3)

The square root of 3 is about 1.7320508

x ~= -6 +/- 3.464101615

x ~= -9.4641  and/or -2.535898  Answer here

+/-  means "plus or minus"; either one.

Last edited by John E. Franklin (2008-01-14 19:08:04)

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