SOLUTION: Milo was asked to find two numbers with the produt of 105 and the sum of 26.She gave the number 15and 7. why was her answer incorrect?

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Question 594575: Milo was asked to find two numbers with the produt of 105 and the sum of 26.She gave the number 15and 7. why was her answer incorrect?
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
Her answer was incorrect because it met only one of the two conditions. The two numbers she proposed do have 105 as their product. However, the sum of her two numbers is 22, not 26 as the problem requires.
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She should have solved the problem by first writing two equations that express the requirements the numbers are to meet. Let x and y represent the two numbers. Then for the product of these two numbers the equation is:
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x * y = 105
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and for the sum of the two numbers the equation is:
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x + y = 26
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One way of solving this system of two equations is by substitution. You can start with the product equation and divide both sides by y to get:
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x = 105/y
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Then in the sum equation you can replace x with its equal 105/y to make the sum equation become:
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105/y + y = 26
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Get rid of the y in the denominator by multiplying both sides (all terms) of this equation by y to get:
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105 + y^2 = 26y
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Put this in standard quadratic form by subtracting 26y from both sides of this equation to make it become:
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105 + y^2 - 26y = 0
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Now rearrange the terms on the left side in descending powers of y to get:
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y^2 - 26y + 105 = 0
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This could be solved by factoring, but let's use the quadratic formula that says for an equation of the standard form:
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ay^2 + by + c = 0
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the solutions will be given by:
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y+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
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by comparing the standard form with the equation we are trying to solve you can see that "a" (the multiplier of the y^2 term) is +1, b (the multiplier of the y term is -26, and c (the constant term) is +105. Substitute these values into the solution equation and it becomes:
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y+=+%28-%28-26%29+%2B-+sqrt%28+%28-26%29%5E2-4%2A1%2A105+%29%29%2F%282%2A1%29+
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First, note that the -(-26) in the numerator is equivalent to +26. And the denominator of 2*1 can be written as just 2. Make these changes and the solution equation becomes:
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y+=+%2826+%2B-+sqrt%28+%28-26%29%5E2-4%2A1%2A105+%29%29%2F2+
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Now work the terms inside the square root sign. Start with squaring -26 to get 676. Next multiply out -4*1*105 to get -420. Substitute these values under the square root sign and you have:
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y+=+%2826+%2B-+sqrt%28+676-420+%29%29%2F2+
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Subtract the 676 - 420 and get 256. This makes the solution equation become:
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y+=+%2826+%2B-+sqrt%28+256+%29%29%2F2+
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Next take the square root of 256. It is 16. Substitute this and the solutions equation is then:
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y+=+%2826+%2B-+16%29%2F2+
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This tells you that there are two solutions for y. The first is:
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y+=+%2826+%2B+16%29%2F2+
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and this simplifies to:
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y+=+%2842%29%2F2+
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which results in y = 21
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The second value for y is given by:
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y+=+%2826+-+16%29%2F2+
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Which tells you that:
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y+=+10%2F2+
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This reduces to y = 5
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So the values that y might be are 21 and 5. If you substitute these two values into either of the original two equations and solve for x you find that when y is 21, x is 5 and when y is 5, x is 21. This makes it fairly clear that the two numbers whose product is 105 and whose sum is 26 are 21 and 5 and these are the correct answer that Milo needed.
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Hope this helps you to understand the problem a little better.
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