document.write( "Question 781819: X^2-4x+c=0. I have to use discriminant. \n" ); document.write( "

Algebra.Com's Answer #476115 by MathLover1(20849) $\"\"$ $\"About$

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$\"x%5E2-4x%2Bc=0\"$ ...where $\"a=1\"$ , $\"b=-4\"$ , and $\"c=c\"$ \r
\n" ); document.write( "\n" ); document.write( "The discriminant $\"D\"$ is the part of the quadratic formula underneath the radical sign: the $\"D=b%5E2+-+4ac\"$ part\r
\n" ); document.write( "\n" ); document.write( " $\"D=b%5E2+-+4ac\"$ ...plug in $\"a=1\"$ , $\"b=-4\"$ , and $\"c=c\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"D=%28-4%29%5E2+-+4%2A1c\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"D=16+-+4c\"$ \r
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\n" ); document.write( "\n" ); document.write( "If $\"D%3E0\"$ , the equation has $\"2\"$ real solutions.
\n" ); document.write( " If $\"D=0\"$ , the equation has $\"1\"$ real solution.
\n" ); document.write( " If $\"D%3C0\"$ , the equation has $\"2\"$ conjugate imaginary solutions.\r
\n" ); document.write( "\n" ); document.write( "so,
\n" ); document.write( " $\"D=16+-+4c\"$ ...set $\"D\"$ less then $\"0\"$ and find $\"c\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"0%3C16+-+4c\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"4c%3C16\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"c%3C4\"$ \r
\n" ); document.write( "\n" ); document.write( "if $\"c%3C4\"$ you will have positive discriminant and that means you will have $\"2\"$ real solutions \r
\n" ); document.write( "\n" ); document.write( " $\"D=16+-+4%2A3\"$
\n" ); document.write( " $\"D=16+-+12\"$
\n" ); document.write( " $\"D=4\"$ => $\"D%3E0\"$ \r
\n" ); document.write( "\n" ); document.write( "check it: take $\"c=3\"$ , plug it in $\"x%5E2-4x%2Bc=0\"$ and solve for $\"x\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"x%5E2-4x%2B3=0\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"x+=+%28-%28-4%29+%2B-+sqrt%28+%28-4%29%5E2-4%2A1%2A3+%29%29%2F%282%2A1%29+\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"x+=+%284+%2B-+sqrt%28+16-12+%29%29%2F2+\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"x+=+%284+%2B-+sqrt%28+4+%29%29%2F2+\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"x+=+%284+%2B-+2%29%2F2+\"$ \r
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\n" ); document.write( "\n" ); document.write( "solutions:\r
\n" ); document.write( "\n" ); document.write( " $\"x+=+%284+%2B2%29%2F2+\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"x+=+6%2F2+\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"x+=+3+\"$ \r
\n" ); document.write( "\n" ); document.write( "or\r
\n" ); document.write( "\n" ); document.write( " $\"x+=+%284+-2%29%2F2+\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"x+=+2%2F2+\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"x+=+1+\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E2-4x%2B3%29+\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"D=16+-+4c\"$ ...set $\"D\"$ equal to $\"0\"$ and find $\"c\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"0=16+-+4c\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"4c=16\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"c=4\"$ \r
\n" ); document.write( "\n" ); document.write( "do same for $\"c=4\"$ to prove that the equation has $\"1\"$ real solution\r
\n" ); document.write( "\n" ); document.write( " $\"D=16+-+4%2A4\"$
\n" ); document.write( " $\"D=16+-+16\"$
\n" ); document.write( " $\"D=0\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E2-4x%2B4%29+\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"D=16+-+4c\"$ ...set $\"D\"$ greater then $\"0\"$ and find $\"c\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"0%3E16+-+4c\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"4c%3E16\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"c%3E4\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "do same for $\"c%3E4\"$ to prove that the equation has $\"2\"$ conjugate imaginary solutions\r
\n" ); document.write( "\n" ); document.write( "take $\"c=5\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"D=16+-+4%2A5\"$
\n" ); document.write( " $\"D=16+-+20\"$
\n" ); document.write( " $\"D=-4\"$ => $\"D%3C0\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E2-4x%2B5%29+\"$ \r
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