For all real numbers x such that x \neq -3, which of the following expressions is equivale… — GRE General Test

For all real numbers $x$ such that $x \neq -3$ , which of the following expressions is equivalent to $\frac{2x^2 - 18}{4x + 12}$ ?

The expression

\frac{x - 3}{2}

is equivalent to the given rational expression.

Factoring the numerator gives

2(x^2 - 9) = 2(x - 3)(x + 3)

and factoring the denominator gives

4(x + 3)

. Canceling the non-zero common terms

2(x + 3)

leaves

\frac{x - 3}{2}

Factor out the greatest common factor from the numerator and denominator.

Numerator:

2x^2 - 18 = 2(x^2 - 9)

; Denominator:

4x + 12 = 4(x + 3)

Factoring out common numerical coefficients simplifies the expression and reveals algebraic patterns.

Apply the difference of squares formula to factor

x^2 - 9

x^2 - 9 = (x - 3)(x + 3)

, so the numerator becomes

2(x - 3)(x + 3)

The algebraic identity

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

allows complete factoring of the numerator.

Cancel common factors shared by the numerator and denominator.

\frac{2(x - 3)(x + 3)}{4(x + 3)} = \frac{2(x - 3)}{4} = \frac{x - 3}{2}.

Since

x \neq -3

, the factor

(x + 3)

is non-zero and can be safely canceled along with reducing the constant ratio

\frac{2}{4}

\frac{1}{2}

Simplifying rational algebraic expressions by factoring common numerical factors and applying the difference of squares identity.

Question