For all non-zero real numbers x, which of the following is equivalent to the expression \f… — ACT

For all non-zero real numbers $x$ , which of the following is equivalent to the expression $\frac{(x+1)^2}{x} - \frac{1}{x}$ ?

$x + 2$ Answer
B
$x$
C
$x^2 + 2x$
D
$x^2 + 2$
E
$x^3 + 2$

Answer

The expression is equivalent to

x + 2

Combining the fractions gives the combined numerator over the denominator. Expanding the binomial in the numerator yields the sum of terms, which simplifies after subtracting one. Factoring out the variable in the numerator and dividing by the same variable simplifies the expression to a first-degree binomial.

Step-by-Step Solution

Combine the two rational terms since they already share a common denominator of

x

\frac{(x+1)^2 - 1}{x}

When subtracting fractions with the same denominator, subtract the numerators and keep the denominator.

Expand the squared binomial

(x+1)^2

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

The square of a binomial is given by

(a+b)^2 = a^2 + 2ab + b^2

Simplify the numerator by combining like terms (

1 - 1 = 0

\frac{x^2 + 2x}{x}

Combine the constant terms in the numerator.

Factor

x

out of each term in the numerator and divide by the denominator

x

x + 2

Since

x \neq 0

, we can cancel the common factor of

x

from both the numerator and the denominator.

Key Concept

Simplifying rational expressions by combining fractions and expanding binomials

Estimated Time:45s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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