document.write( "Question 1071488: Find the range of k in the of equations below if this system has 2 real solutions. Show your work. y=(x-1)^2+3 & y=2x+k \n" ); document.write( "
Algebra.Com's Answer #686450 by KMST(5328)\"\" \"About 
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\"system%28y=2x%2Bk%2Cy=%28x-1%29%5E2%2B3%29\" --> \"system%28k=y-2x%2C%28x-1%29%5E2%2B3%29=2x%2Bk%29\" --> \"system%28k=y-2x%2Cx%5E2-2x%2B1%2B3=2x%2Bk%29\" --> \"system%28k=y-2x%2Cx%5E2-2x%2B4-2x-k=0%29\" --> \"system%28k=y-2x%2Cx%5E2-4x%2B4-k=0%29\"
\n" ); document.write( "If we chose a real \"k\" so that \"x%5E2-4x%2B4-k=0\" has 2 real solutions for \"x\" ,
\n" ); document.write( "we will have the corresponding real solutions for \"y=2x%2Bk\" .
\n" ); document.write( "For a quadratic equation \"ax%5E2%2Bbx%2Bc=0\" to have 2 real solutions
\n" ); document.write( "(which could both be the same)
\n" ); document.write( "it must be \"b%5E2-4ac%3E=0\" .
\n" ); document.write( "For the two real solutions to be different we need \"b%5E2-4ac%3E0\" .
\n" ); document.write( "So, in the case of \"x%5E2-4x%2B4-k=0\" ,
\n" ); document.write( "where \"a=1\", \"b=-4\" and \"c=4-k\" ,
\n" ); document.write( "to have 2 real solutions (which could both be the same) we must have
\n" ); document.write( "\"%28-4%29%5E2-4%2A1%2A%284-k%29%3E=0\" --> \"16-4%284-k%29%3E=0\" --> \"16-16%2B4k%3E=0\" --> \"4k%3E=0\" --> \"highlight%28k%3E=0%29\" .
\n" ); document.write( "As an interval it would be \"%22%5B+0+%2C%22\"\"infinity\"\"%22%29%22\" .
\n" ); document.write( "
\n" ); document.write( "If we wanted to have two different real solutions,
\n" ); document.write( "we would need to have \"highlight%28k%3E0%29\" ,
\n" ); document.write( "and in interval notation that would be \"%22%28+0+%2C%22\"\"infinity\"\"%22%29%22\" .
\n" ); document.write( "
\n" ); document.write( "IF YOU WERE STUDYING CALCULUS,
\n" ); document.write( "the answer would be obvious, if somewhat hard to explain the reasoning.
\n" ); document.write( "You would know that the slope of the tangent to \"y=%28x-1%29%5E2%2B3%29\"
\n" ); document.write( "is \"dy%2Fdx=2%28x-1%29\" ,
\n" ); document.write( "and since the slope of \"y=2x%2Bk\" is \"2\" ,
\n" ); document.write( "the slope of both functions would be the same when
\n" ); document.write( "\"2%28x-1%29=2\" <--> \"x-1=1\" <--> \"x=2\" .
\n" ); document.write( "That point could be the point of tangency if
\n" ); document.write( "both functions have the same \"y\" value for \"x=2\" .
\n" ); document.write( "That would happen for a real \"k\" such that at \"x=2\" <--> \"x-1=1\"
\n" ); document.write( "\"%28x-1%29%5E2%2B3=2x%2Bk\" , meaning that
\n" ); document.write( "\"1%5E2%2B3=2%2A2%2Bk\"
\n" ); document.write( "\"4=4%2Bk\" <---> \"k=0\" .
\n" ); document.write( "That point of tangency would be (2,4), with \"y=2%2A2%2B0=4\" .
\n" ); document.write( "For \"x=2\" <--> \"x-1=1\" ,
\n" ); document.write( "the function \"y=%28x-1%29%5E2%2B3%29\" has \"y%282%29=4\" at \"x=2\" .
\n" ); document.write( "For \"k=0\" both functions would have the \"y%282%29=4\" at \"x=2\" \"graph%28300%2C300%2C-1%2C4%2C-2%2C8%2C-3%2C-4%2C%28x-1%29%5E2%2B3%2C2x%29\"
\n" ); document.write( "If \"green%28k%29%3E0\" function \"y=2x%2Bgreen%28k%29\" will have \"y%282%29=2%2A2%2Bgreen%28k%29=4%2Bgreen%28k%29%3E4\" , and will be passing through a point inside the parabola, crossing it twice:
\n" ); document.write( "If \"red%28k%29%3C0\" function \"y=2x%2Bred%28k%29\" will have \"y%282%29=2%2A2%2Bred%28k%29=4%2Bred%28k%29%3C4\" , and will be below the point (2,4) where the slope of \"y=%28x-1%29%5E2%2B3%29\" is \"2\" ,
\n" ); document.write( "and since the slope of \"y=%28x-1%29%5E2%2B3%29\" increase with increasing x,
\n" ); document.write( "the function \"y=2x%2Bred%28k%29\" , with its constant \"slope=2\" will never be able to catch up.
\n" ); document.write( "\"graph%28300%2C300%2C-1%2C4%2C-2%2C8%2C2x-1.2+%2C2x%2B0.8%2C%28x-1%29%5E2%2B3%2C2x%29\" \n" ); document.write( "
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