28) (x-2)^2 + (x-3) = 2x^2

Solution:

• Expand the squared term: \( (x-2)^2 = x^2 - 4x + 4 \)
• Substitute into the equation: \( x^2 - 4x + 4 + x - 3 = 2x^2 \)
• Combine like terms on the left side: \( x^2 - 3x + 1 = 2x^2 \)
• Rearrange into a quadratic equation: \( x^2 + 3x - 1 = 0 \)
• Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where a=1, b=3, c=-1.
• Calculate the discriminant: \( \Delta = b^2 - 4ac = (3)^2 - 4(1)(-1) = 9 + 4 = 13 \).
• The solutions are: \( x = \frac{-3 \pm \sqrt{13}}{2} \)

Final Answer: \( x = \frac{-3 \pm \sqrt{13}}{2} \)