document.write( "Question 253227: state all the values of k that would make the following have imaginary roots: x^2-9x+k=0\r
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Algebra.Com's Answer #185496 by drk(1908) $\"\"$ $\"About$

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We are given the original equation as:
\n" ); document.write( "(i) $\"x%5E2-9x%2Bk=0\"$
\n" ); document.write( "--
\n" ); document.write( "First, we should solve for x using the quadratic formula:
\n" ); document.write( "(ii) $\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"$
\n" ); document.write( "replacing a, b, and c with 1, -9, and k, we get
\n" ); document.write( "(iii) $\"x+=+%289+%2B-+sqrt%28+81-4%2Ak+%29%29%2F%282%29+\"$
\n" ); document.write( "Now we want imaginary answers, which means the discriminant must < 0. So we get
\n" ); document.write( " $\"81+-+4k+%3C+0\"$
\n" ); document.write( "Solving for k, we get
\n" ); document.write( " $\"k+%3E+20.25\"$
\n" ); document.write( "check:
\n" ); document.write( "Let k = 21.
\n" ); document.write( "From (i) we get
\n" ); document.write( " $\"x%5E2+-9x+%2B+21+=+0\"$
\n" ); document.write( "from (ii), we get
\n" ); document.write( " $\"x+=+%289+%2B-+sqrt%28+81-84+%29%29%2F%282%29+\"$
\n" ); document.write( "and we can see that we get a negative square root. This tells us that our answer is correct.\r
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