document.write( "Question 1209938: Let f be a function such that
\n" ); document.write( "f(x) + f(2x + y) + 5xy = f(4x - y) - x^2 + 5xy - 8x + 17y + 1
\n" ); document.write( "for all real numbers x and y. Find f(10).
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Algebra.Com's Answer #851112 by CPhill(1959)\"\" \"About 
You can put this solution on YOUR website!
Let's solve this functional equation problem step by step.\r
\n" ); document.write( "\n" ); document.write( "**1. Simplify the equation:**\r
\n" ); document.write( "\n" ); document.write( "f(x) + f(2x + y) + 5xy = f(4x - y) - x² + 5xy - 8x + 17y + 1\r
\n" ); document.write( "\n" ); document.write( "Subtract 5xy from both sides:\r
\n" ); document.write( "\n" ); document.write( "f(x) + f(2x + y) = f(4x - y) - x² - 8x + 17y + 1\r
\n" ); document.write( "\n" ); document.write( "**2. Strategically substitute values for x and y:**\r
\n" ); document.write( "\n" ); document.write( "* **Let x = 0:**\r
\n" ); document.write( "\n" ); document.write( "f(0) + f(y) = f(-y) + 17y + 1\r
\n" ); document.write( "\n" ); document.write( "* **Let y = 0:**\r
\n" ); document.write( "\n" ); document.write( "f(x) + f(2x) = f(4x) - x² - 8x + 1\r
\n" ); document.write( "\n" ); document.write( "* **Let y = 2x:**\r
\n" ); document.write( "\n" ); document.write( "f(x) + f(4x) = f(0) - x² + 34x + 1\r
\n" ); document.write( "\n" ); document.write( "* **Let y = 4x:**\r
\n" ); document.write( "\n" ); document.write( "f(x) + f(6x) = f(0) - x² + 68x + 1\r
\n" ); document.write( "\n" ); document.write( "* **Let y = -2x:**\r
\n" ); document.write( "\n" ); document.write( "f(x) + f(0) = f(6x) - x² - 8x - 34x + 1
\n" ); document.write( "f(x) + f(0) = f(6x) - x² - 42x + 1\r
\n" ); document.write( "\n" ); document.write( "**3. Combine equations:**\r
\n" ); document.write( "\n" ); document.write( "From the equations with y=4x and y=-2x:\r
\n" ); document.write( "\n" ); document.write( "f(x) + f(6x) = f(0) - x² + 68x + 1
\n" ); document.write( "f(x) + f(0) = f(6x) - x² - 42x + 1\r
\n" ); document.write( "\n" ); document.write( "Subtract the second equation from the first:\r
\n" ); document.write( "\n" ); document.write( "f(6x) - f(0) = f(0) - x² + 68x + 1
\n" ); document.write( "f(6x) = 2f(0) - x² + 68x + 1\r
\n" ); document.write( "\n" ); document.write( "Subtract f(x) from f(x) + f(2x) = f(4x) - x² - 8x + 1 and f(x) + f(4x) = f(0) - x² + 34x + 1\r
\n" ); document.write( "\n" ); document.write( "f(2x) - f(4x) = f(4x) - f(0) - 42x\r
\n" ); document.write( "\n" ); document.write( "f(2x) -2f(4x)+f(0)=-42x\r
\n" ); document.write( "\n" ); document.write( "**4. Test for a quadratic form:**\r
\n" ); document.write( "\n" ); document.write( "Let f(x) = ax² + bx + c.\r
\n" ); document.write( "\n" ); document.write( "* f(0) = c\r
\n" ); document.write( "\n" ); document.write( "Substitute f(x) into the original equation:\r
\n" ); document.write( "\n" ); document.write( "ax² + bx + c + a(2x+y)² + b(2x+y) + c + 5xy = a(4x-y)² + b(4x-y) + c - x² - 8x + 17y + 1\r
\n" ); document.write( "\n" ); document.write( "ax² + bx + c + a(4x² + 4xy + y²) + 2bx + by + c + 5xy = a(16x² - 8xy + y²) + 4bx - by + c - x² - 8x + 17y + 1\r
\n" ); document.write( "\n" ); document.write( "ax² + bx + c + 4ax² + 4axy + ay² + 2bx + by + c + 5xy = 16ax² - 8axy + ay² + 4bx - by + c - x² - 8x + 17y + 1\r
\n" ); document.write( "\n" ); document.write( "(5a)x² + (3b)x + 2c + (4a+5)xy + ay² +by = (16a-1)x² + (4b-8)x + c + (-8a)xy + ay² + (-b+17)y + 1\r
\n" ); document.write( "\n" ); document.write( "Equating coefficients:\r
\n" ); document.write( "\n" ); document.write( "* x²: 5a = 16a - 1 => 11a = 1 => a = 1/11
\n" ); document.write( "* x: 3b = 4b - 8 => b = 8
\n" ); document.write( "* c: 2c = c + 1 => c = 1
\n" ); document.write( "* xy: 4a + 5 = -8a => 12a = -5 => a = -5/12
\n" ); document.write( "* y²: a = a
\n" ); document.write( "* y: b = -b + 17 => 2b = 17 => b = 17/2\r
\n" ); document.write( "\n" ); document.write( "Since the coefficients don't agree, the form f(x) = ax² + bx + c does not work.\r
\n" ); document.write( "\n" ); document.write( "**5. Try f(x) = ax + b:**\r
\n" ); document.write( "\n" ); document.write( "ax + b + a(2x+y) + b + 5xy = a(4x-y) + b - x² - 8x + 17y + 1\r
\n" ); document.write( "\n" ); document.write( "ax + b + 2ax + ay + b + 5xy = 4ax - ay + b - x² - 8x + 17y + 1\r
\n" ); document.write( "\n" ); document.write( "3ax + 2b + ay + 5xy = 4ax - ay + b - x² - 8x + 17y + 1\r
\n" ); document.write( "\n" ); document.write( "* x² = 0 => 0 = -1 (contradiction)\r
\n" ); document.write( "\n" ); document.write( "**6. Try f(x) = ax^2 + bx + c, y=0**\r
\n" ); document.write( "\n" ); document.write( "f(x) + f(2x) = f(4x) - x^2 - 8x + 1
\n" ); document.write( "ax^2+bx+c+a(2x)^2+b(2x)+c = a(4x)^2+b(4x)+c-x^2-8x+1
\n" ); document.write( "5ax^2+3bx+2c = 16ax^2+4bx+c-x^2-8x+1
\n" ); document.write( "0=11ax^2+bx-c-x^2-8x+1
\n" ); document.write( "0=(11a-1)x^2+(b-8)x+(1-c)\r
\n" ); document.write( "\n" ); document.write( "11a-1=0 => a=1/11
\n" ); document.write( "b-8=0 => b=8
\n" ); document.write( "1-c=0 => c=1\r
\n" ); document.write( "\n" ); document.write( "f(x) = x^2/11 + 8x + 1\r
\n" ); document.write( "\n" ); document.write( "f(10) = 100/11 + 80 + 1 = 100/11 + 81 = 100/11 + 891/11 = 991/11
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