document.write( "Question 1202224: Can you help me write a quadratic equation whose zeros are 8 and 2?
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Algebra.Com's Answer #836899 by math_tutor2020(3817)\"\" \"About 
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\n" ); document.write( "Answer: x^2 - 10x + 16 = 0
\n" ); document.write( "Other equations are possible\r
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\n" ); document.write( "\n" ); document.write( "Explanation:\r
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\n" ); document.write( "\n" ); document.write( "The term \"zero\" of a function is the same as a root.
\n" ); document.write( "For real numbers, the root is the x intercept.\r
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\n" ); document.write( "\n" ); document.write( "The given roots are 8 and 2
\n" ); document.write( "x = 8 leads to x-8 = 0, so (x-8) is one factor
\n" ); document.write( "x = 2 leads to (x-2) being the other factor\r
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\n" ); document.write( "\n" ); document.write( "We then need to expand out (x-8)(x-2)\r
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\n" ); document.write( "\n" ); document.write( "We could use the FOIL rule to expand it out, but I'll use the box method\r
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\n" ); document.write( "\n" ); document.write( "First place the terms along the left and top edge like so
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x-8
x
-2
\r
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\n" ); document.write( "\n" ); document.write( "Then fill each box with the product of the headers
\n" ); document.write( "Example: top left corner is x^2 because x*x = x^2
\n" ); document.write( "Another example: bottom right corner is 16 because -2*(-8) = 16\r
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\n" ); document.write( "\n" ); document.write( "This is what the table would look like when everything is filled in
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x-8
xx^2-8x
-2-2x16
\r
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\n" ); document.write( "\n" ); document.write( "Then we add up the terms.
\n" ); document.write( "Combine like terms if possible
\n" ); document.write( "x^2 + (-8x) + (-2x) + 16
\n" ); document.write( "x^2 - 10x + 16\r
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\n" ); document.write( "\n" ); document.write( "Therefore, (x-8)(x-2) = x^2 - 10x + 16\r
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\n" ); document.write( "\n" ); document.write( "The equation
\n" ); document.write( "x^2 - 10x + 16 = 0
\n" ); document.write( "leads to the roots x = 8 and x = 2
\n" ); document.write( "Other equations lead to these roots because we can scale up or down the equation. For instance, triple both sides to go from x^2-10x+16 = 0 to 3x^2-30x+48 = 0. Both of those equations have the same roots.\r
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\n" ); document.write( "\n" ); document.write( "Check:
\n" ); document.write( "Let's try x = 8
\n" ); document.write( "x^2 - 10x + 16 = 0
\n" ); document.write( "8^2 - 10*8 + 16 = 0
\n" ); document.write( "64 - 10*8 + 16 = 0
\n" ); document.write( "64 - 80 + 16 = 0
\n" ); document.write( "-16 + 16 = 0
\n" ); document.write( "0 = 0
\n" ); document.write( "This confirms x = 8 as a root
\n" ); document.write( "I'll let you try x = 2.
\n" ); document.write( "The goal is to get the left hand side to be zero.\r
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\n" ); document.write( "\n" ); document.write( "Another way to verify is to use a graphing tool like Desmos as shown here
\n" ); document.write( "https://www.desmos.com/calculator/gcibt9azif
\n" ); document.write( "The parabola has x intercepts 2 and 8 where the graph crosses the x axis.
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