document.write( "Question 1209759: Find the sum of solutions x ∈ R of the equation
\n" ); document.write( "[(3cos2x + cos³2x)/(cos⁶x - sin⁶x)] = x³ - x² + 6 \n" ); document.write( "
Algebra.Com's Answer #850191 by CPhill(1959)\"\" \"About 
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Let's analyze the given equation step-by-step:\r
\n" ); document.write( "\n" ); document.write( "**1. Simplify the Trigonometric Expression:**\r
\n" ); document.write( "\n" ); document.write( "* **Numerator:**
\n" ); document.write( " * 3cos(2x) + cos³(2x) = cos(2x)(3 + cos²(2x))\r
\n" ); document.write( "\n" ); document.write( "* **Denominator:**
\n" ); document.write( " * cos⁶(x) - sin⁶(x) = (cos²(x))³ - (sin²(x))³
\n" ); document.write( " * Using a³ - b³ = (a - b)(a² + ab + b²):
\n" ); document.write( " * (cos²(x) - sin²(x))(cos⁴(x) + cos²(x)sin²(x) + sin⁴(x))
\n" ); document.write( " * cos²(x) - sin²(x) = cos(2x)
\n" ); document.write( " * cos⁴(x) + sin⁴(x) = (cos²(x) + sin²(x))² - 2cos²(x)sin²(x) = 1 - 2cos²(x)sin²(x)
\n" ); document.write( " * cos⁶(x) - sin⁶(x) = cos(2x)(1 - 2cos²(x)sin²(x) + cos²(x)sin²(x))
\n" ); document.write( " * cos⁶(x) - sin⁶(x) = cos(2x)(1 - cos²(x)sin²(x))
\n" ); document.write( " * cos²(x)sin²(x) = (1/4)(2sin(x)cos(x))² = (1/4)sin²(2x)
\n" ); document.write( " * cos⁶(x) - sin⁶(x) = cos(2x)(1 - (1/4)sin²(2x))\r
\n" ); document.write( "\n" ); document.write( "* **Substitute into the Fraction:**
\n" ); document.write( " * [cos(2x)(3 + cos²(2x))] / [cos(2x)(1 - (1/4)sin²(2x))]
\n" ); document.write( " * (3 + cos²(2x)) / (1 - (1/4)sin²(2x))\r
\n" ); document.write( "\n" ); document.write( "* **Use cos²(2x) = 1 - sin²(2x):**
\n" ); document.write( " * (3 + 1 - sin²(2x)) / (1 - (1/4)sin²(2x))
\n" ); document.write( " * (4 - sin²(2x)) / (1 - (1/4)sin²(2x))
\n" ); document.write( " * Let y = sin²(2x):
\n" ); document.write( " * (4 - y) / (1 - y/4) = (4 - y) / ((4 - y)/4) = 4\r
\n" ); document.write( "\n" ); document.write( "**2. Simplify the Equation:**\r
\n" ); document.write( "\n" ); document.write( "* The equation becomes: 4 = x³ - x² + 6
\n" ); document.write( "* x³ - x² + 2 = 0\r
\n" ); document.write( "\n" ); document.write( "**3. Find the Roots:**\r
\n" ); document.write( "\n" ); document.write( "* Let's try x = -1: (-1)³ - (-1)² + 2 = -1 - 1 + 2 = 0
\n" ); document.write( "* Therefore, x = -1 is a root.
\n" ); document.write( "* We can perform polynomial division to find the other factors:
\n" ); document.write( " * (x³ - x² + 2) / (x + 1) = x² - 2x + 2
\n" ); document.write( "* The quadratic x² - 2x + 2 has discriminant (-2)² - 4(1)(2) = 4 - 8 = -4, which is negative. Therefore, it has no real roots.\r
\n" ); document.write( "\n" ); document.write( "**4. Find the Sum of Real Solutions:**\r
\n" ); document.write( "\n" ); document.write( "* The only real solution is x = -1.\r
\n" ); document.write( "\n" ); document.write( "**Final Answer:**\r
\n" ); document.write( "\n" ); document.write( "The sum of real solutions is -1.
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