SOLUTION: what two numbers have a product of 48 and a sum of 24.

Question 85927: what two numbers have a product of 48 and a sum of 24.
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
Would it help you to know that this problem does not have a "nice" solution?
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Call the two unknown numbers x and y.
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The problem tells you that the product of the two numbers is 48. Put this in equation form:
.

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The problem also tells you that their sum is 24. Put this in equation form also:
.

.
Let's solve these two equations by variable elimination. Solve one of the two equations
for one of the variables in terms of the other. For example, take the first equation
(the product equation) and solve for x in terms of y. Do this by dividing both sides of the
equation by y. When you do that you get:
.

.
On the left side the y in the denominator cancels the y in the numerator and the equation
becomes:
.

.
Now take the right side of this equation and substitute it into the second equation
(the sum equation) in place of x since the right side is equal to x:
.

.
You can get rid of the denominator by multiplying both sides (all terms) by y. When you
do that the equation becomes:
.

.
Cancel the y in the denominator of the first term with the y in the numerator, and also
recognize that y*y is y-squared. The equation then becomes:
.

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Subtract 24y from both sides of the equation and it becomes:
.

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Arrange the terms in descending powers of y and you have the standard quadratic equation form:
.

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Use the quadratic formula to solve for y. The quadratic formula says that for a quadratic
equation of the form:
.

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the solution for y is:
.

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By comparing the quadratic form with the equation you are trying to solve you can see
that a = 1, b = -24, and c = 48. Substitute these values into the equation for y and you get:
.

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Notice that -(-24) equals + 24, (-24)^2 = 576, and -(4*1*48) = -192, and (2*1) = 2. Substitute
these values into the equation for y and you get:
.

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Now note that (576 - 192) = 384 and the square root of 384 is 19.59591794. Substitute
this value for the radical term and the equation for y becomes:
.

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This generates two possible values for y. When you have the + sign, the value of y becomes:
.

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You know that the sum of x and y must equal 24. So if y = 21.79795897, then x must equal:
.

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So one possible set of answers is y = 21.79795897 and x = 2.20204103.
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You can check this set by adding x and y and make sure they add up to 24 and then multiply
x and y and make sure the product is 48.
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This answer set was based on the use of a + sign preceding the radical in the equation
for y. Now you need to consider the - sign in front of the radical. When you do you get:
.

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Again, the term in the radical is simplifies to 384, and the square root of 384 is 19.59591794.
Substitute this into the equation for y and this time it becomes:
.

.
Find x by subtracting this answer from 24 and you get:
.

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So this time the answer set is y = 2.20204103 and x = 21.79795897.
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But when you compare the answer two sets of answers, you find that one of the numbers
is 21.79795897 and the other number is 2.20204103.
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So the answer to this problem is just that ... the two numbers are 21.79795897 and 2.20204103.
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As I said at the start, the answers are not very "nice" and the only reason they don't have
more than 8 places after the decimal point is that is the number of places my calculator
will handle. If it would compute more places, there would be more than 8 decimal positions.
.
I'm not sure this will help, but it may convince you that a problem that looks so ordinary
at the beginning can sometimes result in answers that look very odd, indeed.
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