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Question 110565: Will someone please help me solve this problem? The problem states: find two positive integers that differ by four and whose product is 45.
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Let x and y represent the two integers you are looking for.
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The problem tells you that the difference between the two integers is 4. Therefore, you
can write the equation:
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x - y = 4 <=== equation 1
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The problem also tells you that the product of the two integers is 45. This leads to another
equation:
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x*y = 45 <=== equation 2
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Now return to equation 1 and solve it for x. You can do this by getting rid of the -y on the left
side. Do this by adding y to both sides of the equation to get:
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x = y + 4
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Since the right side of this equation is equivalent to x, you can now go to equation 2
and substitute y + 4 for x to make equation 2 become:
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(y + 4)*y = 45
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Multiply out the right side to get:
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y^2 + 4y = 45
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Subtract 45 from both sides results in the "standard quadratic form"
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y^2 + 4y - 45 = 0
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The left side of this equation factors into:
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(y + 9)*(y - 5) = 0
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This equation will be true if either factor on the left side is zero because if either factor
is zero the left side will involve a multiplication by zero. Therefore, the left side will be
equal to zero and will therefore equal the right side.
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Set each of the factors equal to zero ... first:
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y + 9 = 0
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Solve for y by adding -9 to both sides to get y = -9. But y = -9 cannot be a solution to
this problem because the problem requires that both answers be a positive integer.
So discard -9 as one of the answers.
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Then try the the other factor ... set y - 5 equal to zero and solve for y.
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y - 5 = 0
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Add +5 to both sides and the answer becomes y = +5. This is the first integer we are looking for.
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Next, since from equation 1 you know x - y = 4. Substitute +5 for y and you have:
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x - (+5) = 4
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This reduces to:
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x - 5 = 4
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Solve for x by adding +5 to both sides to get:
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x = 9
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This is the second integer.
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So the two integers are 9 and 5. As the problem requires, the difference between these integers
if 4 and the product of 9 and 5 is 45.
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Hope this helps you to understand the problem.
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