document.write( "Question 1209411: Find all solutions of the equation |x^2 - 30x - 1| = 30 + |6x - 2| + |x^2 - 20x - 4|. \n" ); document.write( "

Algebra.Com's Answer #848776 by KMST(5328) $\"\"$ $\"About$

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Before graphing calculators, it would have been a more complicated
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\n" ); document.write( "The right hand is greater than 30 except for all values of $\"x\"$
\n" ); document.write( " $\"30%2Babs%286x-2%29%2Babs%286x-2%29%3E=30\"$ for all of real values because
\n" ); document.write( " $\"abs%286x-2%29%3E=0\"$ for all real numbers,
\n" ); document.write( "with $\"abs%286x-2%29=6x-2%3E=0\"$ for $\"x%3E=1%2F3\"$ , and $\"abs%286x-2%29=-6x%2B2\"$ otherwise.
\n" ); document.write( " $\"abs%28x%5E2+-+20x+-+4%29%3E=0\"$ for
\n" ); document.write( " $\"x%3C=10-sqrt%28104%29=approximately\"$ $\"-0.198039027\"$ and $\"x%3E=10%2Bsqrt%28104%29=approximately\"$ $\"20.198039027\"$ .
\n" ); document.write( " $\"x%5E2+-+20x+-+4%3C0\"$ and $\"abs%28x%5E2+-+20x+-+4%29%3E0\"$ for
\n" ); document.write( "for $\"x\"$ such that $\"10-sqrt%28104%29%3Cx%3C10%2Bsqrt%28104%29\"$
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\n" ); document.write( "For the left hand side, $\"x%5E2+-+30x+-+1+=0\"$ , has as solutions $\"x=15+%2B-+sqrt%28226%29\"$
\n" ); document.write( "For values of $\"x%3C=15+-+sqrt%28226%29=approximately\"$ $\"-0.033296378\"$ and $\"x%3E=15+%2B+sqrt%28226%29=approximately\"$ $\"30.033296378\"$ ,
\n" ); document.write( " $\"x%5E2+-+30x+-+1%3E=0\"$ and $\"abs%28x%5E2+-+30x+-+1%29=x%5E2+-+30x+-+1\"$ .
\n" ); document.write( "For $\"15+-+sqrt%28226%29%3Cx%3C15+%2B+sqrt%28226%29\"$ $\"abs%28x%5E2+-+30x+-+1%29=-x%5E2+%2B+30x+%2B+1\"$
\n" ); document.write( "Those limiting values are approximately $\"-0.033296378\"$ and $\"30.033296378\"$ .\r
\n" ); document.write( "\n" ); document.write( "
\n" ); document.write( "For $\"x%3C=10-sqrt%28104%29=approximately\"$ $\"-0.198039027\"$ , $\"6x-2%3C0\"$ , while $\"x%5E2+-+30x+-+1+%3E0\"$ and $\"x%5E2+-+20x+-+4%3E=0\"$
\n" ); document.write( "and the equation turns into
\n" ); document.write( " $\"x%5E2+-+30x+-+1+=+-6x%2B2+%2B+x%5E2+-+20x+-+4\"$
\n" ); document.write( " $\"-30x+%2B+6x+%2B+20x+=+30+%2B+2+-+4+%2B+1\"$
\n" ); document.write( " $\"-4x+=+29\"$ --> $\"x=29%2F%28-4%29=highlight%28-7.25%29\"$
\n" ); document.write( "
\n" ); document.write( "For $\"-1%3Cx%3C1\"$ $\"x%5E2+-+30x+-+1\"$ is decreasing from $\"30\"$ at $\"x=-1\"$ to a minimum 0f $\"-1\"$ at $\"x=0\"$ and then increasing to $\"30\"$ at $\"x=1\"$ and $\"abs%28x%5E2+-+30x+-+1%29\"$ is less than the value of $\"30%2Babs%286x-2%29%2Babs%28x%5E2+-+20x+-+4%29%3E=30\"$
\n" ); document.write( " =
\n" ); document.write( "For $\"1%3C=x%3C=10%2Bsqrt%28104%29\"$ , $\"abs%286x-2%29=6x-2%3E0\"$ , $\"x%5E2-20x-4%3C=0\"$ , and $\"x%5E2-30x-1%3C0\"$ ,
\n" ); document.write( "so $\"abs%28x%5E2-20x-4%29=-x%5E2%2B20x%2B4%29\"$ and $\"abs%28x%5E2-30x-1%29=-x%5E2%2B30x%2B1\"$
\n" ); document.write( "The equation turns into
\n" ); document.write( " $\"-x%5E2%2B30x%2B1=30%2B6x-2%2B-x%5E2%2B20x%2B4\"$
\n" ); document.write( " $\"30x-6x-20x=30-2%2B4-1\"$
\n" ); document.write( " $\"4x=31\"$ --> $\"x=31%2F4=highlight%287.75%29\"$
\n" ); document.write( "
\n" ); document.write( "For $\"10%2Bsqrt%28104%29%3Cx%3C15%2Bsqrt%28226%29\"$ $\"abs%286x-2%29=6x-2%3E0\"$ while
\n" ); document.write( " $\"abs%28x%5E2-20x-4%29=x%5E2-20x-4%3E0\"$ and $\"x%5E2-30x-1%3C0\"$ , so $\"abs%28x%5E2-30x-1%29=-x%5E2%2B30x%2B1\"$
\n" ); document.write( "turning the equation into $\"-x%5E2%2B30x%2B1=30%2B6x-2%2Bx%5E2-20x-4\"$
\n" ); document.write( " $\"0=x%5E2-30x-1%2B30%2B6x-2%2Bx%5E2-20x-4\"$
\n" ); document.write( " $\"0=2x%5E2-44x%2B23\"$ --> $\"x=%2844+%2B-+sqrt%281752%29%29%2F4=11+%2B-+sqrt%28438%29%2F2\"$
\n" ); document.write( " $\"x=11-sqrt%28438%29%2F2=approximately\"$ $\"0.536\"$ does not comply with $\"10%2Bsqrt%28104%29%3Cx%3C15%2Bsqrt%28226%29\"$
\n" ); document.write( " $\"x=highlight%2811%2Bsqrt%28438%29%2F2%29=approximately\"$ $\"21.464225\"$ is a solution between $\"10%2Bsqrt%28104%29=approximately\"$ $\"20.198039027\"$ and $\"15%2Bsqrt%28226%29=approximately\"$ $\"30.033296378\"$
\n" ); document.write( "
\n" ); document.write( "For $\"x%3E=15%2Bsqrt%28226%29\"$ $\"abs%286x-2%29=6x-2%3E0\"$ , $\"abs%28x%5E2-20x-4%29=x%5E2-20x-4%3E0\"$ , and $\"abs%28x%5E2-30x-1%29=x%5E2-30x-1%3E=0\"$ ,
\n" ); document.write( "turning the equation into $\"x%5E2-30x-1=30%2B6x-2%2Bx%5E2-20x-4\"$
\n" ); document.write( " $\"-30x-6x%2B20x=30-2-4%2B1\"$
\n" ); document.write( " $\"-16x=25\"$ --> $\"x=25%2F%28-16%29=-1.5625\"$ is not a solution complying with $\"x%3E=15%2Bsqrt%28226%29=approximately\"$ $\"30.033296378\"$ \n" ); document.write( "

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