Problem-solving
8.7 Algebra Expressions
Show all working. Partial marks are given for method.
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1**Equivalent expressions check.** Determine whether each pair of expressions is equivalent, by expanding/simplifying: (a) $3(x + 4)$ and $3x + 12$. (b) $(x + 2)^2$ and $x^2 + 4$. (c) $2(x - 3) - (x - 5)$ and $x - 1$. (d) $\dfrac{6x}{2}$ and $3x$.
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2**Substitution with mixed signs.** Calculate the value of each expression for $x = -3, y = 2$: (a) $2x + 3y$ (b) $x^2 - y^2$ (c) $\dfrac{x + y}{x - y}$ (d) $(2x - y)^2$
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3**Factorise practice.** Factorise each expression fully. (a) $6x + 12$ (b) $4x^2 - 8x$ (c) $12 a^2 b - 18 a b^2$ (d) $5(x - 1) + y(x - 1)$
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4**Algebraic fractions.** Simplify each: (a) $\dfrac{8a^2 b}{4 a b^2}$ (b) $\dfrac{6x^2 - 9x}{3x}$ (c) $\dfrac{x}{2} + \dfrac{2x}{5}$ (d) $\dfrac{x - 3}{4} - \dfrac{x + 1}{6}$
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5**Songs in terms of $x$ (department Algebra Feb 2022).** Pete has three times as many songs as Sabrina. Sabrina has 120 fewer songs than Elena. (a) If Elena has $x$ songs, write expressions for (i) Sabrina's songs and (ii) Pete's songs. (b) Pete has 300 songs. Write and solve an equation to find Elena's count.
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6**Rectangle expressions.** A rectangle has dimensions $(2x + 1)$ and $(x + 3)$. (a) Write expressions for perimeter and area. (b) Expand the area expression. (c) When $x = 4$, find numerical values for perimeter and area.
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7**Two-bracket expansion.** Expand and simplify each: (a) $(x + 5)(x + 3)$ (b) $(x - 4)(x + 7)$ (c) $(2x - 1)(x + 3)$ (d) $(x + 2)^2$
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8**Pete & Sabrina extended.** Pete has 3 times as many songs as Sabrina. Sabrina has $d$ fewer songs than Elena. If Pete has $P$ songs, express Elena's number of songs in terms of $P$ and $d$.
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9**Algebra in geometry.** A triangle has angles $(2x)°$, $(3x + 10)°$, $(x - 10)°$. (a) Set up an equation in $x$ and solve. (b) Find the three angles.
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10**Patterns in algebraic identities.** Verify (by expansion) that for any $a$ and $b$: (a) $(a + b)^2 = a^2 + 2ab + b^2$ (b) $(a - b)^2 = a^2 - 2ab + b^2$ (c) $(a + b)(a - b) = a^2 - b^2$ (d) Use (a) to compute $52^2$ from $50^2 = 2500$.
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11**Comparing fractions algebraically.** Simplify each and decide which is larger when $x > 0$: (a) $\dfrac{x + 6}{2}$ vs $\dfrac{2x + 6}{3}$ (b) Find the value of $x$ where they are equal.
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12**Repeating-decimal investigation.** Recurring decimals can be turned into fractions algebraically. Let $x = 0.\overline{ab}$ (two-digit repeating block). (a) Show that $100x - x = ab$ and hence $x = \tfrac{ab}{99}$. (b) Use this to convert $0.\overline{36}$ to a fraction in simplest form. (c) Convert $0.\overline{142857}$ — what fraction do you recognise?
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