27) (x-6)^2 + (x+8) = 2x^2

Solution:

• Expand the squared term: \( (x-6)^2 = x^2 - 12x + 36 \)
• Substitute into the equation: \( x^2 - 12x + 36 + x + 8 = 2x^2 \)
• Combine like terms: \( x^2 - 11x + 44 = 2x^2 \)
• Rearrange into a quadratic equation: \( x^2 + 11x - 44 = 0 \)
• Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where a=1, b=11, c=-44.
• Calculate the discriminant: \( \Delta = b^2 - 4ac = (11)^2 - 4(1)(-44) = 121 + 176 = 297 \).
• The solutions are: \( x = \frac{-11 \pm \sqrt{297}}{2} \)

Final Answer: \( x = \frac{-11 \pm \sqrt{297}}{2} \)