document.write( "Question 727151: draw an example of a quadratic equation that has two, one and no solutions. \n" ); document.write( "

Algebra.Com's Answer #444990 by MathLover1(20849) $\"\"$ $\"About$

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How to determine the number of solutions.
\n" ); document.write( "Let's look at the quadratic formula:
\n" ); document.write( " $\"x+=+%28-b+%2B-+sqrt%28b%5E2+-+4ac%29%29%2F2a\"$
\n" ); document.write( "The key is the expression in the square root: $\"b%5E2+-+4ac\"$ .
\n" ); document.write( "In general there are three cases:

$\"b%5E2+-4ac\"$ is positive.
\n" ); document.write( "The square root of a positive number is also some positive number.
\n" ); document.write( "So in the numerator of the quadratic formula we will get two values:
\n" ); document.write( "( $\"-b\"$ + the square root) and ( $\"-b\"$ - the square root).
\n" ); document.write( "So when $\"b%5E2+-+4ac+%3E+0\"$ we get $\"two\"$ $\"+solutions\"$ .
\n" ); document.write( "
$\"b%5E2+-4ac\"$ is zero.
\n" ); document.write( "The square root of zero is zero. So in the numerator we get ( $\"-b+%2B+0\"$ ) and ( $\"-b+-+0\"$ ).
\n" ); document.write( "But both of these are equal to $\"-b\"$ !
\n" ); document.write( "So when $\"b%5E2+-+4ac+=+0\"$ we only get $\"one\"$ $\"+solution\"$ .
\n" ); document.write( "
$\"b%5E2+-4ac\"$ is negative. What is the square root of a negative number? What can we square and get a negative number as an answer? Answer: $\"Nothing\"$ .
\n" ); document.write( "You cannot square any Real number and get a negative. So, when $\"b%5E2+-4ac+%3C+0\"$ there are $\"no\"$ $\"+solutions\"$ .
\n" ); document.write( "

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\n" ); document.write( "\n" ); document.write( "example:
\n" ); document.write( "1.
\n" ); document.write( " $\"x%5E2+%2B+8x+-+4+=+0\"$ ... $\"8%5E2+-+4%2A1%2A%28-4%29+=+0\"$ which is $\"64+%2B16+=+80\"$ ..=> $\"b%5E2+-+4ac+%3E+0\"$ ; so, we have $\"two\"$ $\"+solutions\"$ \r
\n" ); document.write( "\n" ); document.write( "2.\r
\n" ); document.write( "\n" ); document.write( " $\"x%5E2+%2B+4x+%2B+4+=+0\"$ ... $\"4%5E2+-+4%2A1%2A%284%29+=+0\"$ which is $\"16-16+=+0\"$ ..=> $\"b%5E2+-+4ac+=+0\"$ ; so, we have $\"one\"$ $\"+solution\"$ \r
\n" ); document.write( "\n" ); document.write( "3.\r
\n" ); document.write( "\n" ); document.write( " $\"x%5E2+%2B+4x+%2B+8+=+0\"$ ... $\"4%5E2+-+4%2A1%2A%288%29+=+0\"$ which is $\"16-32+=+-6\"$ ..=> $\"b%5E2+-+4ac+%3C0\"$ ; so, we have $\"no\"$ $\"+solution\"$ \r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

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