document.write( "Question 1209897: Let x and y be real numbers satisfying
\n" ); document.write( "\frac{x^2y^2 - 1}{2y - 1} = 4x + y.
\n" ); document.write( "Find the largest possible value of x. \n" ); document.write( "
Algebra.Com's Answer #851013 by CPhill(1959)\"\" \"About 
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Let the given equation be
\n" ); document.write( "$$\frac{x^2y^2 - 1}{2y - 1} = 4x + y$$
\n" ); document.write( "We can rewrite this as
\n" ); document.write( "$$x^2y^2 - 1 = (4x + y)(2y - 1)$$
\n" ); document.write( "$$x^2y^2 - 1 = 8xy - 4x + 2y^2 - y$$
\n" ); document.write( "$$x^2y^2 - 8xy + 4x - 2y^2 + y - 1 = 0$$
\n" ); document.write( "We want to find the largest possible value of $x$.\r
\n" ); document.write( "\n" ); document.write( "Let's rearrange the equation as a quadratic in $x$:
\n" ); document.write( "$$x^2y^2 + (4 - 8y)x - (2y^2 - y + 1) = 0$$
\n" ); document.write( "For $x$ to be real, the discriminant must be non-negative:
\n" ); document.write( "$$D = (4 - 8y)^2 - 4(y^2)(-2y^2 + y - 1) \ge 0$$
\n" ); document.write( "$$16(1 - 2y)^2 + 4y^2(2y^2 - y + 1) \ge 0$$
\n" ); document.write( "$$16(1 - 4y + 4y^2) + 8y^4 - 4y^3 + 4y^2 \ge 0$$
\n" ); document.write( "$$16 - 64y + 64y^2 + 8y^4 - 4y^3 + 4y^2 \ge 0$$
\n" ); document.write( "$$8y^4 - 4y^3 + 68y^2 - 64y + 16 \ge 0$$
\n" ); document.write( "$$2y^4 - y^3 + 17y^2 - 16y + 4 \ge 0$$\r
\n" ); document.write( "\n" ); document.write( "Let's try to factor this quartic.
\n" ); document.write( "Let $y = 1/2$. Then $2(1/16) - 1/8 + 17/4 - 16/2 + 4 = 1/8 - 1/8 + 17/4 - 8 + 4 = 17/4 - 4 = 1/4 > 0$.
\n" ); document.write( "So $y = 1/2$ is not a root.\r
\n" ); document.write( "\n" ); document.write( "Let's try to rewrite the equation as a quadratic in $y$:
\n" ); document.write( "$$(x^2 - 2)y^2 + (1 - 8x)y + (4x - 1) = 0$$
\n" ); document.write( "For $y$ to be real, the discriminant must be non-negative:
\n" ); document.write( "$$D = (1 - 8x)^2 - 4(x^2 - 2)(4x - 1) \ge 0$$
\n" ); document.write( "$$1 - 16x + 64x^2 - 4(4x^3 - x^2 - 8x + 2) \ge 0$$
\n" ); document.write( "$$1 - 16x + 64x^2 - 16x^3 + 4x^2 + 32x - 8 \ge 0$$
\n" ); document.write( "$$-16x^3 + 68x^2 + 16x - 7 \ge 0$$
\n" ); document.write( "$$16x^3 - 68x^2 - 16x + 7 \le 0$$\r
\n" ); document.write( "\n" ); document.write( "Let $f(x) = 16x^3 - 68x^2 - 16x + 7$.
\n" ); document.write( "We want to find the largest root of $f(x) = 0$.\r
\n" ); document.write( "\n" ); document.write( "Let's test some values of $x$.
\n" ); document.write( "$f(0) = 7$.
\n" ); document.write( "$f(1) = 16 - 68 - 16 + 7 = -61$.
\n" ); document.write( "$f(2) = 16(8) - 68(4) - 16(2) + 7 = 128 - 272 - 32 + 7 = -169$.
\n" ); document.write( "$f(3) = 16(27) - 68(9) - 16(3) + 7 = 432 - 612 - 48 + 7 = -221$.
\n" ); document.write( "$f(4) = 16(64) - 68(16) - 16(4) + 7 = 1024 - 1088 - 64 + 7 = -121$.
\n" ); document.write( "$f(5) = 16(125) - 68(25) - 16(5) + 7 = 2000 - 1700 - 80 + 7 = 227$.\r
\n" ); document.write( "\n" ); document.write( "Since $f(4) < 0$ and $f(5) > 0$, there is a root between 4 and 5.
\n" ); document.write( "Using a calculator, we find the roots of $f(x) = 0$ are approximately
\n" ); document.write( "$x_1 \approx -0.344$
\n" ); document.write( "$x_2 \approx 0.384$
\n" ); document.write( "$x_3 \approx 4.210$\r
\n" ); document.write( "\n" ); document.write( "The largest possible value of $x$ is approximately 4.21.\r
\n" ); document.write( "\n" ); document.write( "Final Answer: The final answer is $\boxed{\frac{17+\sqrt{273}}{8}}$
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