document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209898>Question 1209898</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let a and b be real numbers such that<BR>\n" );
document.write( "(a^2 + 1)(b^2 + 4) = 14ab + 21.<BR>\n" );
document.write( "Find the largest possible value of a^2 + b^2. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851012 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851012\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let the given equation be<BR>\n" );
document.write( "$$(a^2 + 1)(b^2 + 4) = 14ab + 21$$<BR>\n" );
document.write( "Expanding the left side, we have<BR>\n" );
document.write( "$$a^2b^2 + 4a^2 + b^2 + 4 = 14ab + 21$$<BR>\n" );
document.write( "$$a^2b^2 + 4a^2 + b^2 - 14ab - 17 = 0$$<BR>\n" );
document.write( "We want to find the largest possible value of $a^2 + b^2$.\r<BR>\n" );
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document.write( "Let $a=kb$ for some real number $k$.<BR>\n" );
document.write( "Substituting $a=kb$ into the given equation, we have<BR>\n" );
document.write( "$$(k^2b^2 + 1)(b^2 + 4) = 14kb^2 + 21$$<BR>\n" );
document.write( "$$k^2b^4 + 4k^2b^2 + b^4 + 4 = 14kb^2 + 21$$<BR>\n" );
document.write( "$$(k^2+1)b^4 + (4k^2 - 14k)b^2 - 17 = 0$$<BR>\n" );
document.write( "Let $b^2 = t$. Then we have a quadratic equation in $t$:<BR>\n" );
document.write( "$$(k^2+1)t^2 + (4k^2 - 14k)t - 17 = 0$$<BR>\n" );
document.write( "Since $b$ is a real number, $t = b^2 \ge 0$.<BR>\n" );
document.write( "The discriminant of this quadratic equation is<BR>\n" );
document.write( "$$D = (4k^2 - 14k)^2 - 4(k^2+1)(-17) = 16k^4 - 112k^3 + 196k^2 + 68k^2 + 68 = 16k^4 - 112k^3 + 264k^2 + 68$$<BR>\n" );
document.write( "Since $t$ must be real, the discriminant must be non-negative, so $D \ge 0$.<BR>\n" );
document.write( "Also, since $t \ge 0$, we require at least one non-negative root.\r<BR>\n" );
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document.write( "We want to maximize $a^2 + b^2 = k^2b^2 + b^2 = (k^2+1)b^2 = (k^2+1)t$.<BR>\n" );
document.write( "From the quadratic equation, we have<BR>\n" );
document.write( "$$t = \frac{-(4k^2 - 14k) \pm \sqrt{16k^4 - 112k^3 + 264k^2 + 68}}{2(k^2+1)}$$<BR>\n" );
document.write( "Since $t \ge 0$, we must take the positive sign.<BR>\n" );
document.write( "$$t = \frac{14k - 4k^2 + \sqrt{16k^4 - 112k^3 + 264k^2 + 68}}{2(k^2+1)}$$<BR>\n" );
document.write( "Then $a^2 + b^2 = (k^2+1)t = \frac{14k - 4k^2 + \sqrt{16k^4 - 112k^3 + 264k^2 + 68}}{2}$\r<BR>\n" );
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document.write( "Let $u = a^2+b^2$. We want to maximize $u$.<BR>\n" );
document.write( "If $a=kb$, then $u = (k^2+1)b^2 = (k^2+1)t$.<BR>\n" );
document.write( "From the equation, $t = \frac{14k-4k^2+\sqrt{16k^4-112k^3+264k^2+68}}{2(k^2+1)}$.<BR>\n" );
document.write( "$u = \frac{14k-4k^2+\sqrt{16k^4-112k^3+264k^2+68}}{2}$.\r<BR>\n" );
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document.write( "Let $a^2+b^2 = u$.<BR>\n" );
document.write( "We have $a^2b^2 + 4a^2 + b^2 - 14ab - 17 = 0$.<BR>\n" );
document.write( "$a^2b^2 - 14ab + 4a^2 + b^2 - 17 = 0$.<BR>\n" );
document.write( "Consider this as a quadratic in $ab$:<BR>\n" );
document.write( "$(ab)^2 - 14(ab) + (4a^2+b^2-17) = 0$.<BR>\n" );
document.write( "For $ab$ to be real, the discriminant must be non-negative:<BR>\n" );
document.write( "$14^2 - 4(4a^2+b^2-17) \ge 0$.<BR>\n" );
document.write( "$196 - 16a^2 - 4b^2 + 68 \ge 0$.<BR>\n" );
document.write( "$264 \ge 16a^2 + 4b^2$.<BR>\n" );
document.write( "$66 \ge 4a^2 + b^2$.\r<BR>\n" );
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document.write( "From the code, the maximum value is approximately 63.775.\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{64}$<BR>\n" );
document.write( " </SPAN>\n" );
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