You can put this solution on YOUR website! Hi, there--
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Find 2 numbers whose sum is 12 and whose product is the maxium possible value.
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Let x be the first number. The sum of the two numbers is 12 so 12-x is an expression for the second number.
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Let y be the product of the two numbers.
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Then an equation that represents this relationship is
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Expand the right hand side of the equation.
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Re-order the terms.
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This is a quadratic equation; its graph is a parabola. Since the leading coefficient is negative the vertex represents the maximum value of the equation.
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If the vertex of a parabola is the point (h,k), then h=-b/2a. In the formula, a is the coefficient of the x-squared terms, and b is the coefficient of the x-term when the quadratic equation is in general form [y=ax^2+bx+c].
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In our equation, a is -1 and b is 12.
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The equation h=6 means that the x-coordinate of the vertex is 6. Substitute 6 for x in the quadratic equation to find the y-coordinate.
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The y-coordinate of the vertex is 36, so the vertex is (6,36).
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Now we need to interpret our answer in terms of this problem. Recall that x is the first number. The first number is 6. The second number is also 6 since 12-x=12-6=6. The y-coordinate is the product of the two numbers, 36.
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So the two numbers are 6 and 6.
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Feel free to email via gmail if the solution is unclear.
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Ms.Figgy
math.in.the.vortex@gmail.com
