If x > 2 and satisfies the equation x - \frac{8}{x^2 - 2x} = 5 - \frac{4}{x - 2} what is t… — SAT

If $x > 2$ and satisfies the equation

x - \frac{8}{x^2 - 2x} = 5 - \frac{4}{x - 2}

what is the value of $x$ ?

Answer: 4

The correct answer is 4.

To solve the rational equation, we first determine the domain restrictions:

x \neq 0

and

x \neq 2

. We then clear the denominators by multiplying the entire equation by the least common denominator

x(x - 2)

, which simplifies the equation to the cubic form

x^3 - 7x^2 + 14x - 8 = 0

. Factoring the cubic polynomial yields

(x - 1)(x - 2)(x - 4) = 0

. The potential solutions are

x = 1

x = 2

, and

x = 4

. Since

x = 2

is an extraneous solution and the constraint requires

x > 2

x = 4

is the only valid solution.

Determine the domain restrictions of the rational terms in the equation.

x \neq 0

and

x \neq 2

Division by zero is undefined, so the denominators

x(x-2)

and

x-2

cannot be zero.

Multiply the entire equation by the least common denominator

x(x - 2)

to eliminate the fractions.

x^2(x - 2) - 8 = 5x(x - 2) - 4x

Multiplying by the common denominator converts the rational equation into an equivalent polynomial equation within the defined domain.

Expand the terms and collect them on one side to form a standard polynomial equation.

x^3 - 7x^2 + 14x - 8 = 0

Simplifying the expressions allows us to find the roots of the polynomial.

Factor the cubic polynomial using the rational root theorem or synthetic division.

(x - 1)(x - 2)(x - 4) = 0

Since

x = 1

makes the polynomial zero, we factor out

(x-1)

to get the remaining quadratic factor

x^2 - 6x + 8

, which factors into

(x - 2)(x - 4)

Identify the valid solution based on the domain restriction and the given inequality constraint.

x = 4

The value

x = 2

is extraneous because it makes the original equation undefined. The value

x = 1

is rejected because the problem specifies

x > 2

Solving rational equations by clearing denominators, factoring polynomials, identifying extraneous solutions, and applying inequality constraints.

Question