document.write( "Question 1209123: Let a and b be the roots of x^2 + 7x - 4 = 0. Find (a + 3)/(b + 3) + (b + 3)/(a + 3). \n" ); document.write( "
Algebra.Com's Answer #847712 by math_tutor2020(3816)\"\" \"About 
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\n" ); document.write( "Answer: -33/16\r
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\n" ); document.write( "\n" ); document.write( "Explanation\r
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\n" ); document.write( "\n" ); document.write( "I'll use p,q in place of a,b
\n" ); document.write( "This is because a,b,c are the standard coefficients of the quadratic template \"ax%5E2%2Bbx%2Bc+=+0\".
\n" ); document.write( "In the case of x^2+7x-4 = 0 we have a = 1, b = 7, c = -4.\r
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\n" ); document.write( "\n" ); document.write( "Instead of computing \"%28a+%2B+3%29%2F%28b+%2B+3%29+%2B+%28b+%2B+3%29%2F%28a+%2B+3%29\" the expression I'll evaluate is \"%28p+%2B+3%29%2F%28q+%2B+3%29+%2B+%28q+%2B+3%29%2F%28p+%2B+3%29\"\r
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\n" ); document.write( "\n" ); document.write( "I'll take a slight detour for a moment.\r
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\n" ); document.write( "\n" ); document.write( "From the quadratic version of Vieta's Formulas, we know that:
\n" ); document.write( "p+q = -b/a
\n" ); document.write( "p*q = c/a
\n" ); document.write( "When plugging a = 1, b = 7, and c = -4, we get
\n" ); document.write( "p+q = -b/a = -7/1 = -7
\n" ); document.write( "p*q = c/a = -4/1 = -4\r
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\n" ); document.write( "\n" ); document.write( "In short,
\n" ); document.write( "p+q = -7
\n" ); document.write( "p*q = -4
\n" ); document.write( "Let's call these equation (1) and equation (2) to be used later.\r
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\n" ); document.write( "\n" ); document.write( "Then note the following
\n" ); document.write( "\"%28p%2Bq%29%5E2+=+p%5E2%2B2pq%2Bq%5E2\"\r
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\n" ); document.write( "\n" ); document.write( "\"p%5E2%2Bq%5E2+=+%28p%2Bq%29%5E2-2pq\"\r
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\n" ); document.write( "\n" ); document.write( "\"p%5E2%2Bq%5E2+=+%28-7%29%5E2-2%28-4%29\" Applying equations (1) and (2)\r
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\n" ); document.write( "\n" ); document.write( "\"p%5E2%2Bq%5E2+=+57\" Let's call this equation (3)\r
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\n" ); document.write( "\n" ); document.write( "Let's return to \"%28p+%2B+3%29%2F%28q+%2B+3%29+%2B+%28q+%2B+3%29%2F%28p+%2B+3%29\"
\n" ); document.write( "We'll combine the fractions.
\n" ); document.write( "Recall we need the LCD to do so.\r
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\n" ); document.write( "\n" ); document.write( "\"%28p+%2B+3%29%2F%28q+%2B+3%29+%2B+%28q+%2B+3%29%2F%28p+%2B+3%29\"\r
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\n" ); document.write( "\n" ); document.write( "= \r
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\n" ); document.write( "\n" ); document.write( "= \r
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\n" ); document.write( "\n" ); document.write( "= \"%28p%5E2%2B6p%2B9%2Bq%5E2%2B6q%2B9%29%2F%28pq%2B3p%2B3q%2B9%29\"\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "= \"%28p%5E2%2Bq%5E2%2B6%28p%2Bq%29%2B18%29%2F%28pq%2B3%28p%2Bq%29%2B9%29\"\r
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\n" ); document.write( "\n" ); document.write( "= \r
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\n" ); document.write( "\n" ); document.write( "= Apply equations (1) through (3)\r
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\n" ); document.write( "\n" ); document.write( "= \"%2857%2B6%28-7%29%2B18%29%2F%28-4%2B3%28-7%29%2B9%29\"\r
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\n" ); document.write( "\n" ); document.write( "= \"33%2F%28-16%29\"\r
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\n" ); document.write( "\n" ); document.write( "= \"-33%2F16\"\r
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\n" ); document.write( "\n" ); document.write( "Therefore,
\n" ); document.write( "\"%28p+%2B+3%29%2F%28q+%2B+3%29+%2B+%28q+%2B+3%29%2F%28p+%2B+3%29+=+-33%2F16\"
\n" ); document.write( "where p,q are the roots of \"x%5E2%2B7x-4=0\"\r
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\n" ); document.write( "\n" ); document.write( "To verify, you can use the quadratic formula to solve \"x%5E2%2B7x-4=0\"
\n" ); document.write( "You should get \"p+=+%28-7%2Bsqrt%2865%29%29%2F2\" and \"q+=+%28-7-sqrt%2865%29%29%2F2\" as the two roots.\r
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\n" ); document.write( "\n" ); document.write( "Then plug each value into \"%28p+%2B+3%29%2F%28q+%2B+3%29+%2B+%28q+%2B+3%29%2F%28p+%2B+3%29\" and simplify. \r
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\n" ); document.write( "\n" ); document.write( "I used GeoGebra to verify the answer.
\n" ); document.write( "Here's the link to that calculation
\n" ); document.write( "https://www.geogebra.org/calculator/fwzwpynj
\n" ); document.write( "Let me know if you have any questions.
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