document.write( "Question 1143176: Find the values of a for which the curve y= x^2 never touches the curve y= a-(x-a)^2 \n" ); document.write( "

Algebra.Com's Answer #763956 by MathLover1(20849) $\"\"$ $\"About$

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Find the values of a for which the curve $\"y=+x%5E2\"$ never touches the curve $\"y=+a-%28x-a%29%5E2+\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"y=+x%5E2+\"$
\n" ); document.write( " $\"y=+a-%28x-a%29%5E2\"$
\n" ); document.write( "------------------------\r
\n" ); document.write( "\n" ); document.write( " $\"x%5E2+=a-%28x-a%29%5E2\"$
\n" ); document.write( " $\"x%5E2+=a-%28x%5E2-2xa%2Ba%5E2%29\"$
\n" ); document.write( " $\"x%5E2+%2B%28x%5E2-2xa%2Ba%5E2%29-a=0\"$
\n" ); document.write( " $\"x%5E2+%2Bx%5E2-2xa%2Ba%5E2-a=0\"$
\n" ); document.write( " $\"2x%5E2-2ax%2Ba%5E2-a=0\"$ \r
\n" ); document.write( "\n" ); document.write( "use discriminant $\"D=b%5E2-4ac+\"$
\n" ); document.write( "When $\"a\"$ , $\"+b\"$ , and $\"c\"$ are real numbers, $\"a+%3C%3E+0\"$ and discriminant is $\"negative\"$ , then the roots α and β of the quadratic equation $\"ax%5E2+%2B+bx+%2B+c+=+0+\"$ are unequal and $\"not\"$ real. In this case, we say that the roots are $\"imaginary\"$ .\r
\n" ); document.write( "\n" ); document.write( "in your case discriminant $\"D\"$ is:\r
\n" ); document.write( "\n" ); document.write( " $\"2x%5E2-2ax%2Ba%5E2-a=0\"$ \r
\n" ); document.write( "\n" ); document.write( "since $\"a=2\"$ , $\"+b=-2a\"$ , and $\"c=a%5E2-a\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"D=%28-2a%29%5E2-4%2A2%28a%5E2-a%29\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"D=4a%5E2-8%28a%5E2-a%29\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"D=4a%5E2-8a%5E2%2B8a\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"D=-4a%5E2%2B8a\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"D=-4a%28a-2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "if $\"D%3C0\"$ => $\"-4a%28a-2%29%3C0\"$ \r
\n" ); document.write( "\n" ); document.write( "solutions:\r
\n" ); document.write( "\n" ); document.write( "if $\"-4a+%3E0\"$ => $\"a%3C0\"$
\n" ); document.write( "if $\"%28a-2%29%3E0\"$ => $\"a%3E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "so, your solution is:
\n" ); document.write( " $\"a\"$ is in interval ( $\"-infinity\"$ , $\"0\"$ )
\n" ); document.write( "or
\n" ); document.write( " $\"a\"$ is in interval ( $\"2\"$ , $\"infinity\"$ )\r
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\n" ); document.write( "\n" ); document.write( "check some values:\r
\n" ); document.write( "\n" ); document.write( "let's $\"a=-1\"$ in interval ( $\"-infinity\"$ , $\"0\"$ ) \r
\n" ); document.write( "\n" ); document.write( " $\"y=+x%5E2+\"$
\n" ); document.write( " $\"y=+-1-%28x%2B1%29%5E2\"$ \r
\n" ); document.write( "\n" ); document.write( "graph:\r
\n" ); document.write( "\n" ); document.write( " $\"+graph%28+600%2C+600%2C+-10%2C10%2C+-10%2C+10%2C+x%5E2%2C-1-%28x%2B1%29%5E2%29+\"$ \r
\n" ); document.write( "\n" ); document.write( "so, the curve $\"y=+x%5E2\"$ never touches the curve $\"y=+a-%28x-a%29%5E2\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "let's $\"a=3\"$ in interval ( $\"2\"$ , $\"infinity\"$ )\r
\n" ); document.write( "\n" ); document.write( " $\"y=+x%5E2+\"$
\n" ); document.write( " $\"y=+3-%28x-3%29%5E2\"$ \r
\n" ); document.write( "\n" ); document.write( "graph:\r
\n" ); document.write( "\n" ); document.write( " $\"+graph%28+600%2C+600%2C+-10%2C10%2C+-10%2C+10%2C+x%5E2%2C3-%28x-3%29%5E2%29+\"$ \r
\n" ); document.write( "\n" ); document.write( "so, the curve $\"y=+x%5E2\"$ never touches the curve $\"y=+a-%28x-a%29%5E2\"$ \r
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