Match each algebraic expression to an equivalent form
Equivalent expressions have the same value for every input where the original expression is defined. Simplification is valid when algebraic identities are used correctly and any domain restrictions remain unchanged after cancellations.
Expressions
(1) \(3(2x-5)+4x\)
(2) \(\dfrac{x^2-9}{x-3}\)
(3) \(2(x+1)^2-2(x^2+1)\)
(4) \(\dfrac{6x^2y}{3xy^2}\)
(5) \(\dfrac{\sqrt{50}}{\sqrt{2}}\)
Answer choices
A. \(10x-15\)
B. \(x+3\), with \(x\neq 3\)
C. \(4x\)
D. \(\dfrac{2x}{y}\), with \(x\neq 0\) and \(y\neq 0\)
E. \(5\)
Equivalence and domain
An equivalence statement \(E_1(x)=E_2(x)\) in algebra means equality for every \(x\) in the domain of \(E_1\). When a simplification involves a denominator, the excluded values that make the original denominator zero remain excluded after simplification.
Work and matches
(1) \(3(2x-5)+4x\)
\[ 3(2x-5)+4x = (6x-15)+4x = 10x-15 \]Match: A
(2) \(\dfrac{x^2-9}{x-3}\)
\[ \frac{x^2-9}{x-3} = \frac{(x-3)(x+3)}{x-3} = x+3,\quad x\neq 3 \]Match: B
(3) \(2(x+1)^2-2(x^2+1)\)
\[ 2(x+1)^2-2(x^2+1) = 2(x^2+2x+1) - (2x^2+2) = (2x^2+4x+2) - 2x^2 - 2 = 4x \]Match: C
(4) \(\dfrac{6x^2y}{3xy^2}\)
\[ \frac{6x^2y}{3xy^2} = \frac{6}{3}\cdot\frac{x^2}{x}\cdot\frac{y}{y^2} = 2\cdot x \cdot \frac{1}{y} = \frac{2x}{y},\quad x\neq 0,\ y\neq 0 \]Match: D
(5) \(\dfrac{\sqrt{50}}{\sqrt{2}}\)
\[ \frac{\sqrt{50}}{\sqrt{2}}=\sqrt{\frac{50}{2}}=\sqrt{25}=5 \]Match: E
Final matching table
| Expression | Equivalent choice | Equivalent form | Domain notes |
|---|---|---|---|
| (1) \(3(2x-5)+4x\) | A | \(10x-15\) | All real \(x\) |
| (2) \(\dfrac{x^2-9}{x-3}\) | B | \(x+3\) | \(x\neq 3\) |
| (3) \(2(x+1)^2-2(x^2+1)\) | C | \(4x\) | All real \(x\) |
| (4) \(\dfrac{6x^2y}{3xy^2}\) | D | \(\dfrac{2x}{y}\) | \(x\neq 0,\ y\neq 0\) |
| (5) \(\dfrac{\sqrt{50}}{\sqrt{2}}\) | E | \(5\) | Real values; \(\sqrt{2}\neq 0\) |
Common pitfalls
Cancellation across a fraction changes the appearance of an expression but does not change which inputs are allowed; excluded values from the original denominator remain excluded. Another frequent error is distributing a coefficient incorrectly, such as treating \(3(2x-5)\) as \(6x-5\) instead of \(6x-15\).