document.write( "Question 824722: Please help me with these questions :
\n" ); document.write( "1) The equations ax^2 +bx+c =0 and bx^2 +ax+c=0 have the same roots. Show that a+b+c=0 where a is not =b and c is not =0. \r
\n" ); document.write( "\n" ); document.write( "2) show that the equation (x-k)^ 2 +2x-k has real roots and hence show that 1\n" ); document.write( "
Algebra.Com's Answer #496763 by KMST(5328)\"\" \"About 
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1) If the equations \"ax%5E2%2Bbx%2Bc=0\" and \"bx%5E2%2Bax%2Bc=0\" ,
\n" ); document.write( "where \"a%3C%3Eb\" and \"c%3C%3E0\" , have a common root, that root will satisfy
\n" ); document.write( "\"ax%5E2%2Bbx%2Bc=bx%5E2%2Bax%2Bc\" .
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\n" ); document.write( "Solving:
\n" ); document.write( "\"ax%5E2%2Bbx%2Bc=bx%5E2%2Bax%2Bc\"
\n" ); document.write( "\"ax%5E2%2Bbx%2Bcross%28c%29=bx%5E2%2Bax%2Bcross%28c%29\"
\n" ); document.write( "\"ax%5E2%2Bbx=bx%5E2%2Bax\"
\n" ); document.write( "\"ax%5E2-bx%5E2%2Bbx-ax=0\"
\n" ); document.write( "\"x%28ax-bx%2Bb-a%29=0\"
\n" ); document.write( "If \"x=0\" were a solution of \"ax%5E2%2Bbx%2Bc=0\" , \"c\" would be zero:
\n" ); document.write( "\"a%2A0%5E2%2Bb%2A0%2Bc=0%2B0%2Bc=c=0\" .
\n" ); document.write( "So we know that,
\n" ); document.write( "\"system%28x%3C%3E0%2Cx%28ax-bx%2Bb-a%29=0%29\" --> \"%28ax-bx%2Bb-a%29=0\"
\n" ); document.write( "\"%28ax-bx%2Bb-a%29=0\"
\n" ); document.write( "\"%28a-b%29x%2Bb-a=0\"
\n" ); document.write( "\"%28a-b%29x=a-b\"
\n" ); document.write( "We can divide both sides of the equal sign by \"%28a-b%29%3C%3E0\" , to get
\n" ); document.write( "\"highlight%28x=1%29\".
\n" ); document.write( "(We know that \"%28a-b%29%3C%3E0\" because \"a%3C%3Eb\" ).
\n" ); document.write( "
\n" ); document.write( "Since we have concluded that only \"x=1\" could be a solution to both equations,
\n" ); document.write( "substituting \"1\" for \"x\" must make both equations true.
\n" ); document.write( "Substituting into \"ax%5E2%2Bbx%2Bc=0\" ,
\n" ); document.write( "we get
\n" ); document.write( "\"0=ax%5E2%2Bbx%2Bc=a%2A1%5E2%2Bb%2A1%2Bc\"
\n" ); document.write( "\"0=ax%5E2%2Bbx%2Bc=a%2Bb%2Bc\"
\n" ); document.write( "\"highlight%28a%2Bb%2Bc=0%29\"
\n" ); document.write( "We get the same using the other equation.
\n" ); document.write( "The two equations could be \"x%5E2-3x%2B2=0\" , with roots \"x=1\" and \"x=2\" ,
\n" ); document.write( "and \"-3x%5E2%2Bx%2B2=0\" , with roots \"x=1\" and \"x=-2%2F3\".
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\n" ); document.write( "NOTE:
\n" ); document.write( "The only solution that we found that could be a common solution to both equations was \"x=1\" .
\n" ); document.write( "There must be two more, different solutions, one for each equation.
\n" ); document.write( "It cannot be \"x=1\" as a double solution to both equations,
\n" ); document.write( "because that would require \"a=b\" .
\n" ); document.write( "If \"x=1\" were the only solution to both equations, the equations would be
\n" ); document.write( "\"0=ax%5E2%2Bbx%2Bc=a%28x-1%29%5E2=a%28x%5E2-2x%2B1%29=ax%5E2-2ax%2Bhighlight%28a%29\" , and
\n" ); document.write( "\"0=bx%5E2%2Bax%2Bc=b%28x-1%29%5E2=b%28x%5E2-2x%2B1%29=bx%5E2-2bx%2Bhighlight%28b%29\" .
\n" ); document.write( "The independent term in both should be
\n" ); document.write( "\"c=highlight%28a%29\" and \"c=highlight%28b%29\" .
\n" ); document.write( "That would require \"a=b\" .
\n" ); document.write( "
\n" ); document.write( "2) \"%28x-k%29%5E+2+%2B2x-k=0\"
\n" ); document.write( "\"%28x-k%29%5E+2+%2B2x-k-k=-k\"
\n" ); document.write( "\"%28x-k%29%5E+2+%2B2x-2k=-k\"
\n" ); document.write( "\"%28x-k%29%5E+2+%2B2%28x-k%29=-k\"
\n" ); document.write( "\"%28x-k%29%5E+2+%2B2%28x-k%29%2B1=1-k\"
\n" ); document.write( "\"%28%28x-k%29+%2B1%29%5E+2=1-k\"
\n" ); document.write( "So \"1-k%3E=0\" --> \"1%3E=k\" <--> \"k%3C=1\"
\n" ); document.write( "
\n" ); document.write( "Alternatively,
\n" ); document.write( "\"%28x-k%29%5E+2+%2B2x-k=0\"
\n" ); document.write( "\"x%5E2-2kx%2Bk%5E2%2B2x-k=0\"
\n" ); document.write( "\"x%5E2%2B%282-2k%29x%2Bk%5E2-k=0\"
\n" ); document.write( "For real solutions we need the discriminant to be non negative
\n" ); document.write( "\"%282-2k%29%5E2-4%2A1%2A%28k%5E2-k%29%3E=0\"
\n" ); document.write( "\"4-8k%2B4k%5E2-4k%5E2%2B4k%3E=0\"
\n" ); document.write( "\"4-8k%2B4k%3E=0\"
\n" ); document.write( "\"4%3E=4k\"
\n" ); document.write( "\"1%3E=k\" <--> \"k%3C=1\"
\n" ); document.write( " \n" ); document.write( "
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