25) (x+1) + (x-6) = 2x^2

Solution:

• Expand the terms: \( (x+1) + (x-6) = x + 1 + x - 6 = 2x - 5 \)
• Set up the equation: \( 2x - 5 = 2x^2 \)
• Rearrange into a quadratic equation: \( 2x^2 - 2x + 5 = 0 \)
• Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where a=2, b=-2, c=5.
• Calculate the discriminant: \( \Delta = b^2 - 4ac = (-2)^2 - 4(2)(5) = 4 - 40 = -36 \).
• Since the discriminant is negative, there are no real solutions.

Final Answer: No real solutions