\text{Find the discriminant of the equation}\frac{x-3}{x-4}+\frac{x-6}{x-5}=\frac{5}{2}.

The correct option is -7.

To find the discriminant of the given equation, we first need to simplify the algebraic expression and write it in the standard quadratic form
$$a{x}^2+bx+c=0$$

The given equation is:
$$\frac{x-3}{x-4}+\frac{x-6}{x-5}=\frac{5}{2}$$

Step 1: Find a common denominator for the terms on the left-hand side.
The common denominator is $(x-4)(x-5)$. Combining the fractions gives:
$$\frac{(x-3)(x-5)+(x-6)(x-4)}{(x-4)(x-5)}=\frac{5}{2}$$

Step 2: Expand the numerators and the denominator.
Expanding the numerator terms:
$$(x-3)(x-5)=x^2-8x+15$$
$$(x-6)(x-4)=x^2-10x+24$$

Adding them together in the numerator:
$$(x^2-8x+15)+(x^2-10x+24)=2x^2-18x+39$$

Expanding the denominator:
$$(x-4)(x-5)=x^2-9x+20$$

Substituting these back into our equation:
$$\frac{2x^2-18x+39}{x^2-9x+20}=\frac{5}{2}$$

Step 3: Cross-multiply to eliminate fractions.
$$2(2x^2-18x+39)=5(x^2-9x+20)$$

Distribute the terms:
$$4x^2-36x+78=5x^2-45x+100$$

Step 4: Move all terms to one side to set the equation to zero.
Let us subtract $4x^2-36x+78$ from both sides:
$$0=(5x^2-4x^2)+(-45x+36x)+(100-78)$$
$$x^2-9x+22=0$$

Here, we have a quadratic equation with coefficients:

• a = 1
• b = -9
• c = 22

Step 5: Calculate the discriminant.
The formula for the discriminant (D) of a quadratic equation $a{x}^2+bx+c=0$ is:
$$D=b^2-4ac$$

Substituting the values of a, b, and c:
$$D={(-9)}^2-4\left(1\right)\left(22\right)$$
$$D=81-88$$
$$D=-7$$

Thus, the discriminant of the given equation is -7.