Simplify Rational Equation

Begin with the given rational equation:

$ \frac{x-3}{x-4} + \frac{x-6}{x-5} = \frac{5}{2} $

Combine fractions on the left side using a common denominator:

$ \frac{(x-3)(x-5) + (x-6)(x-4)}{(x-4)(x-5)} = \frac{5}{2} $

Expand and simplify the numerator and denominator:

• Numerator: $ (x^2 - 8x + 15) + (x^2 - 10x + 24) = 2x^2 - 18x + 39 $
• Denominator: $ x^2 - 9x + 20 $

Substitute back into the equation:

$ \frac{2x^2 - 18x + 39}{x^2 - 9x + 20} = \frac{5}{2} $

Cross-multiply:

$ 2(2x^2 - 18x + 39) = 5(x^2 - 9x + 20) $

$ 4x^2 - 36x + 78 = 5x^2 - 45x + 100 $

Rearrange into standard quadratic form $ ax^2 + bx + c = 0 $:

$ 0 = x^2 - 9x + 22 $

Calculate Discriminant

Identify the coefficients $a$, $b$, and $c$ from the quadratic equation $ x^2 - 9x + 22 = 0 $:

• $ a = 1 $
• $ b = -9 $
• $ c = 22 $

Apply the discriminant formula $ \Delta = b^2 - 4ac $:

$ \Delta = (-9)^2 - 4(1)(22) $

$ \Delta = 81 - 88 $

$ \Delta = -7 $