document.write( "Question 1183695: Find the value of K for which y=x+k is a tangent to the curve y=x²+5x+2 \n" ); document.write( "

Algebra.Com's Answer #814130 by ikleyn(52787) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Find the value of K for which y=x+k is a tangent to the curve y=x²+5x+2
\n" ); document.write( "~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "Notice that the slope of the line y = x+k  is equal to  1  (one).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Hence, to solve the problem, we should find a point on the parabola, where the slope is 1,\r\n" );
document.write( "and then to pass the line y = x+k through this point.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The derivative of the quadratic function is  y' = 2x + 5.\r\n" );
document.write( "\r\n" );
document.write( "It is equal to  1  when  2x+5 = 1,  or  x =  =  = -2.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Next, the value of the quadratic function at x= -2 is  (-2)^2 + 5*(-2) + 2 = 4 - 10 + 2 = -4.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now we find the value of \"k\" from this equation  x + k = -4  at  x= -2, i.e.\r\n" );
document.write( "\r\n" );
document.write( "       -2 + k = -4   ---->    k = -4 + 2 = -2.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "ANSWER.  The value of \"k\" is -2.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "                         VISUAL CHECK\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     Plot y = x^2 + 5x + 2 (red),  y = x - 2 (green)\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );