document.write( "Question 840889: Real or imaginary solutions
\n" ); document.write( "x^3=216
\n" ); document.write( "6x=9+x^2
\n" ); document.write( "X^2+3x+3=0
\n" ); document.write( "X^3+125=0
\n" ); document.write( "X^4-24x^2+100=0 \n" ); document.write( "

Algebra.Com's Answer #506622 by KMST(5328) $\"\"$ $\"About$

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$\"x%5E3=216\"$ --> $\"x%5E3-216=0\"$ --> $\"x%5E3-6%5E3=0\"$ --> $\"%28x-6%29%28x%5E2%2B6x%2B6%5E2%29=0\"$ --> $\"%28x-6%29%28x%5E2%2B6x%2B36%29=0\"$
\n" ); document.write( " $\"x=6\"$ , which makes $\"%28x-6%29=0\"$ is the real solution;
\n" ); document.write( " $\"x%5E2%2B6x%2B36=0\"$ , which has no real solution yields two more imaginary ones.
\n" ); document.write( "
\n" ); document.write( " $\"6x=9%2Bx%5E2\"$ --> $\"x%5E2-6x%2B9=0\"$ --> $\"%28x-3%29%5E2=0\"$ has $\"x=3\"$ as a double real solution (no imaginary ones).
\n" ); document.write( "
\n" ); document.write( " $\"x%5E2%2B3x%2B3=0\"$ has no real solutions, just imaginary ones.
\n" ); document.write( "You would know that by calculating the discriminant.
\n" ); document.write( "A quadratic equation, that can be written as
\n" ); document.write( " $\"ax%5E2%2Bbx%2Bc=0\"$ , has solutions that can be calculated using the quadratic formula:
\n" ); document.write( " $\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"$
\n" ); document.write( "The expression $\"b%5E2-4%2Aa%2Ac\"$ is called the discriminant, and its sign will tell you about the solutions. A negative discriminant means no real solutions. A positive one means two (different) real solutions. If the discriminant is zero, there is just one (double) real solution (as happened for $\"x%5E2-6x%2B9=0\"$ above).
\n" ); document.write( "In this case $\"a=1\"$ , $\"b=3\"$ , and $\"c=3\"$ , and
\n" ); document.write( " $\"b%5E2-4%2Aa%2Ac=3%5E2%2A4%2A1%2A3=9-12=-3%3C0\"$ , so no real solutions.
\n" ); document.write( "
\n" ); document.write( " $\"x%5E3%2B125=0\"$ --> $\"x%5E3%2B5%5E3=0\"$ --> $\"%28x%2B5%29%28x%5E2-5x%2B5%5E2%29=0\"$
\n" ); document.write( " $\"x=-5\"$ is the real solution; $\"x%5E2-5x%2B25=0\"$ , which has no real solution yields two more imaginary ones.
\n" ); document.write( "
\n" ); document.write( " $\"x%5E4-24x%5E2%2B100=0\"$ is more complicated.
\n" ); document.write( "You could use the change of variable $\"y=x%5E2\"$ to write the equation as
\n" ); document.write( " $\"y%5E2-24y%2B100=0\"$ , and find that the solutions for $\"y\"$ .
\n" ); document.write( "You could solve for $\"y\"$ by completing the square, or by using the quadratic formula.
\n" ); document.write( "Either way, you will find
\n" ); document.write( " $\"y=12+%2B-+sqrt%2844%29\"$ (or an equivalent expression).
\n" ); document.write( "Since both $\"y\"$ solutions are positive, you will have four real solutions to
\n" ); document.write( " $\"x%5E2=12+%2B-+sqrt%2844%29\"$ , and those are the solutions of
\n" ); document.write( " $\"x%5E4-24x%5E2%2B100=0\"$ .\r
\n" ); document.write( "\n" ); document.write( "so there are two real solutions.
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