Shawazi Morgan
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The sum of two positive numbers is 42 and their product is 425. What are the two numbers.
How would I find the answer to this?
How would I find the answer to this?
The sum of two positive numbers is 42 and their product is 425. What are the two numbers.
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lev888
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The problem almost begs you to set up a system of 2 equations.
HallsofIvy
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That looks straight forward. Call the two numbers "x" and "y".
"The sum of two numbers is 42." So x+ y= 42
"Their product is 425." So xy= 425.
From x+ y= 42, subtracting x from both sides, y= 42- x.
xy= x(42- x)= 42x- x^2= 425.
Adding x^2 to both side and subtracting 42x from both sides,
x^2- 42x= -425.
That's a "quadratic equation". There are many ways to solve such an equation.
I like "completing the square"
Add (42/2)^2= 21^2= 441 to both sides: x^2- 42x+ 441= 441- 425= 16
The point of adding 441 is that it makes the left side a "perfect square":
x^2- 42x+ 441= (x- 21)^2. Of course 4^2= 16 (and so does (-4)^2) so
(x- 21)^2= 4^2= (-4)^2
x- 21= 4 or x- 21= -4.
x= 4+ 21= 25 or x= -4+ 21= 17.
y= 42- x so if x= 25, y= 42- 25= 17 and if x= 17, y= 42- 17= 25.
There are two solutions, x= 25 and y= 17 or x= 17 and y= 25.
Check: if x= 25 and y= 17 then their sum is 25+ 17= 42 and their product is 25(17)= 425.
If x= 17 and y= 25 then their sum is 17+ 25= 42 and their product is 17(25)= 425.
"The sum of two numbers is 42." So x+ y= 42
"Their product is 425." So xy= 425.
From x+ y= 42, subtracting x from both sides, y= 42- x.
xy= x(42- x)= 42x- x^2= 425.
Adding x^2 to both side and subtracting 42x from both sides,
x^2- 42x= -425.
That's a "quadratic equation". There are many ways to solve such an equation.
I like "completing the square"
Add (42/2)^2= 21^2= 441 to both sides: x^2- 42x+ 441= 441- 425= 16
The point of adding 441 is that it makes the left side a "perfect square":
x^2- 42x+ 441= (x- 21)^2. Of course 4^2= 16 (and so does (-4)^2) so
(x- 21)^2= 4^2= (-4)^2
x- 21= 4 or x- 21= -4.
x= 4+ 21= 25 or x= -4+ 21= 17.
y= 42- x so if x= 25, y= 42- 25= 17 and if x= 17, y= 42- 17= 25.
There are two solutions, x= 25 and y= 17 or x= 17 and y= 25.
Check: if x= 25 and y= 17 then their sum is 25+ 17= 42 and their product is 25(17)= 425.
If x= 17 and y= 25 then their sum is 17+ 25= 42 and their product is 17(25)= 425.
Dr.Peterson
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If you had reason to think the numbers might be integers, you could just look at factor pairs for 425 and see if any of them worked. They are 1*425, 5*85, 17*25. One of those has the right sum, so you have the answer.
But the other suggested methods take less guessing, and will work even if the numbers are not integers.
Yes, it's worth taking the time to think before rushing in! Most probably they were integers, so a very good strategy to start with.
In fact if you factor (using your calculator) you get: [MATH]5^2 \times 17[/MATH]The two numbers are staring at you. Nothing to be done.