If x is a real number, what is the sum of all distinct real solutions to the rational equa… — ACT

If $x$ is a real number, what is the sum of all distinct real solutions to the rational equation $\frac{3}{(x-2)^2} - \frac{2}{x-2} = 1$ ?

The sum of all distinct real solutions is 2.

The correct answer is obtained by multiplying the equation by the least common denominator,

(x-2)^2

, which yields the quadratic equation

x^2 - 2x - 3 = 0

. Factoring this equation gives two solutions:

3

and

-1

. Since neither value makes the original denominator equal to zero, both are valid solutions. Summing these values gives a result of

2

Identify the domain constraints of the rational equation.

The denominator

x-2

cannot be zero, so the domain constraint is

x \neq 2

Division by zero is undefined.

Eliminate the denominators by multiplying the entire equation by the least common denominator, which is

(x-2)^2

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

This transforms the rational equation into a polynomial equation.

Expand both sides of the equation and combine like terms.

3 - 2x + 4 = x^2 - 4x + 4 \implies 7 - 2x = x^2 - 4x + 4

Standard algebraic expansion simplifies the equation into a quadratic form.

Rearrange the quadratic equation into standard form

ax^2 + bx + c = 0

x^2 - 2x - 3 = 0

Setting the equation to zero allows us to solve for

x

by factoring.

Factor the quadratic equation.

(x-3)(x+1) = 0 \implies x = 3

x = -1

Factoring finds the values of

x

that satisfy the quadratic equation.

Verify if the potential solutions satisfy the domain constraint and calculate their sum.

Both

3

and

-1

are within the domain (

x \neq 2

). The sum is

3 + (-1) = 2

Extraneous solutions must be discarded before calculating the final sum.

Solving rational equations by clearing denominators and verifying domain constraints

Question