document.write( "Question 629643: if the roots of x^2-ax+b=0 are real & differ by a quantity which is less than c(c>0),prove that b lies between (1/4)(a^2-c^2)& (1/4)a^2 \n" ); document.write( "

Algebra.Com's Answer #396439 by jsmallt9(3758) $\"\"$ $\"About$

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Using the Quadratic Formula on your equation we get:
\n" ); document.write( " $\"x+=+%28-%28-a%29+%2B-+sqrt%28%28-a%29%5E2-4%281%29%28b%29%29%29%2F2%281%29\"$
\n" ); document.write( "which simplifies as follows:
\n" ); document.write( " $\"x+=+%28-%28-a%29+%2B-+sqrt%28a%5E2-4%281%29%28b%29%29%29%2F2%281%29\"$
\n" ); document.write( " $\"x+=+%28-%28-a%29+%2B-+sqrt%28a%5E2-4b%29%29%2F2%281%29\"$
\n" ); document.write( " $\"x+=+%28a+%2B-+sqrt%28a%5E2-4b%29%29%2F2\"$
\n" ); document.write( "which is short for:
\n" ); document.write( " $\"x+=+%28a+%2B+sqrt%28a%5E2-4b%29%29%2F2\"$ or $\"x+=+%28a+-+sqrt%28a%5E2-4b%29%29%2F2\"$

\n" ); document.write( "The difference of these is:
\n" ); document.write( " $\"%28a+%2B+sqrt%28a%5E2-4b%29%29%2F2+-+%28a+-+sqrt%28a%5E2-4b%29%29%2F2\"$
\n" ); document.write( " $\"%282sqrt%28a%5E2-4b%29%29%2F2\"$
\n" ); document.write( " $\"sqrt%28a%5E2-4b%29\"$

\n" ); document.write( "We are told that this difference is less than c, and as is true of any square root, it is not negative. So
\n" ); document.write( " $\"sqrt%28a%5E2-4b%29+%3C+c\"$ and $\"sqrt%28a%5E2-4b%29+%3E=+0\"$
\n" ); document.write( "Squaring both sides of both we get:
\n" ); document.write( " $\"a%5E2-4b+%3C+c%5E2\"$ and $\"a%5E2-4b+%3E=+0\"$
\n" ); document.write( "Now we solve both for b:
\n" ); document.write( " $\"a%5E2-c%5E2+%3C+4b\"$ and $\"a%5E2+%3E=+4b\"$
\n" ); document.write( " $\"%281%2F4%29%28a%5E2-c%5E2%29+%3C+b\"$ and $\"%281%2F4%29a%5E2+%3E=+b\"$

\n" ); document.write( "P.S. To me, \"between p and q\" means \"between p and q but not equal to p or q\". But I don't think there is any way to eliminate the \"or equal to\" part of $\"%281%2F4%29a%5E2+%3E=+b\"$ . So the largest possible b would be equal to $\"%281%2F4%29a%5E2\"$ not between that and $\"%281%2F4%29%28a%5E2-c%5E2%29\"$ . \n" ); document.write( "

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