An equation is shown below. \frac{x+2}{x-1} - \frac{4}{x} = \frac{4}{x^2 - x} What is the … — SAT

An equation is shown below.

\frac{x+2}{x-1} - \frac{4}{x} = \frac{4}{x^2 - x}

What is the real solution to the equation above?

Cevap: 2

The only real solution to the equation is

2

To solve the rational equation, we multiply all terms by the common denominator

x(x-1)

, assuming

x \neq 0

and

x \neq 1

. This results in the equation

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

. Expanding and simplifying this gives

x^2 + 2x - 4x + 4 = 4

, which simplifies to

x^2 - 2x = 0

. Factoring the quadratic expression yields

x(x-2) = 0

, which gives the potential solutions

x = 0

and

x = 2

. However, substituting

x = 0

back into the original equation causes a division by zero. Therefore,

x = 0

is an extraneous solution, and the only valid real solution is

2

Find the common denominator for the terms in the rational equation.

The common denominator is

x(x-1) = x^2 - x

, which requires

x \neq 0

and

x \neq 1

Multiplying by the common denominator allows us to eliminate the fractions.

Multiply every term in the equation by the common denominator

x(x-1)

and simplify.

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

This clears the denominators and converts the rational equation into a polynomial equation.

Expand and simplify the resulting equation to standard quadratic form.

x^2 + 2x - 4x + 4 = 4

, which simplifies to

x^2 - 2x = 0

Grouping like terms is necessary to solve the quadratic equation.

Factor the quadratic equation to determine the potential solutions.

x(x-2) = 0

, giving potential solutions of

x = 0

and

x = 2

Factoring allows us to find the roots of the quadratic expression.

Check the potential solutions in the original equation to identify any extraneous solutions.

x = 0

is extraneous because it leads to division by zero. Thus, the only real solution is

x = 2

Solutions that make any denominator in the original equation equal to zero must be excluded.

Solving rational equations and identifying extraneous solutions

Soru