LAWS OF ALGEBRA

QUESTION 1 [6 marks] — Criterion A Medium

Expand each perfect square:

a. $(x+6)^2$

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$$x^2+12x+36$$

b. $(x-4)^2$

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$$x^2-8x+16$$

c. $(3x+2)^2$

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$$9x^2+12x+4$$

QUESTION 2 [5 marks] — Criterion B Medium

Investigate the pattern in $(x+a)^2$ compared to $(x-a)^2$.

a. Expand $(x+5)^2$ and $(x-5)^2$.

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$(x+5)^2=x^2+10x+25$. $(x-5)^2=x^2-10x+25$.

b. Compare the two expansions. What stays the same, and what changes?

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The $x^2$ term and the constant term ($25$) are IDENTICAL in both. Only the sign of the middle term ($+10x$ vs $-10x$) changes.

c. State the general rule for $(x-a)^2$ compared to $(x+a)^2 = x^2+2ax+a^2$.

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$$(x-a)^2 = x^2-2ax+a^2$$ (only the middle term's sign flips)

QUESTION 3 [5 marks] — Criterion C Medium

A student expands $(x+7)^2$ as $x^2+49$ (squaring each term separately, forgetting the middle term).

a. Explain the error using the definition $(x+7)^2=(x+7)(x+7)$, and give the correct expansion.

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$(x+7)^2$ means $(x+7)(x+7)$, which by FOIL gives FOUR terms ($x^2$, $7x$, $7x$, $49$), not just two. The student incorrectly assumed squaring 'distributes' over addition (i.e. $(a+b)^2=a^2+b^2$), which is a common but false shortcut. Correct: $x^2+14x+49$.

b. Verify using $x=3$: evaluate both $(x+7)^2$ directly and the student's (incorrect) expansion.

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$(3+7)^2=10^2=100$. Student's version: $3^2+49=9+49=58\ne100$ — confirms the error.

QUESTION 4 [5 marks] — Criterion D Medium

A square swimming pool has side length $(x+3)$ m, surrounded by a uniform paved border making the OUTER edge of the paving $(x+3+4)=(x+7)$ m per side (since the border is 2 m wide on each side).

a. Write and expand an expression for the area of the pool.

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$$(x+3)^2=x^2+6x+9 \text{ m}^2$$

b. Write and expand an expression for the TOTAL area (pool plus paving), then find the area of JUST the paving (border) in simplified form.

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Total area $=(x+7)^2=x^2+14x+49$. Paving area $=(x^2+14x+49)-(x^2+6x+9)=8x+40$.

QUESTION 5 [7 marks] — Criterion A Medium

Expand each perfect square:

a. $(x+11)^2$

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$$x^2+22x+121$$

b. $(x-7)^2$

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$$x^2-14x+49$$

c. $(4x+1)^2$

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$$16x^2+8x+1$$

QUESTION 6 [5 marks] — Criterion B Hard

Investigate whether $(x+y)^2$ can EVER equal $x^2+y^2$ (i.e. whether the 'cross term' $2xy$ can ever vanish).

a. Expand $(x+y)^2$ fully, identifying the cross term.

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$$(x+y)^2 = x^2+2xy+y^2$$ — the cross term is $2xy$.

b. For the cross term $2xy$ to equal exactly 0 (making $(x+y)^2=x^2+y^2$ true), what must be true about $x$ or $y$? Test your conclusion with a specific example.

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$2xy=0$ requires EITHER $x=0$ OR $y=0$ (or both) — since if neither is zero, their product can't be zero. Test: $x=0,y=7$: $(0+7)^2=49$, and $0^2+7^2=49$ — matches ?, confirming $(x+y)^2=x^2+y^2$ ONLY when at least one of $x,y$ is zero.

QUESTION 7 [6 marks] — Criterion B Hard

Investigate the pattern in $(x+1)^2-(x-1)^2$ compared to $(x+2)^2-(x-2)^2$, looking for a general rule.

a. Fully expand and simplify $(x+1)^2-(x-1)^2$.

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$(x^2+2x+1)-(x^2-2x+1)=4x$.

b. Now do the same for $(x+2)^2-(x-2)^2$, and state the general pattern for $(x+a)^2-(x-a)^2$.

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$(x^2+4x+4)-(x^2-4x+4)=8x$. General pattern: $$(x+a)^2-(x-a)^2=4ax$$ (verify: with $a=1$, $4(1)x=4x$ ?; with $a=2$, $4(2)x=8x$ ?).

QUESTION 8 [5 marks] — Criterion C Medium

A student expands $(x-9)^2$ as $x^2-81$ (treating it like a difference of squares instead of a perfect square).

a. Explain the error, clarifying the difference between $(x-9)^2$ and $(x+9)(x-9)$.

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$(x-9)^2$ means $(x-9)(x-9)$ — the SAME bracket multiplied by itself, which expands to $x^2-18x+81$ (including a middle term). The student's answer, $x^2-81$, is actually the expansion of the DIFFERENT expression $(x+9)(x-9)$ (a difference of squares, two DIFFERENT brackets) — confusing a perfect square with a difference of squares is a common but significant error.

b. Verify the correct expansion of $(x-9)^2$ using $x=10$.

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$(10-9)^2=1^2=1$. Correct formula $x^2-18x+81$ at $x=10$: $100-180+81=1$ ? (student's wrong version $x^2-81$ at $x=10$ gives $19\ne1$).

QUESTION 9 [5 marks] — Criterion C Hard

A classmate says: 'since $(x+5)^2$ has a PLUS in the brackets, the middle term of its expansion must also be positive — and since $(x-5)^2$ has a MINUS, its middle term must be negative.' They conclude this rule always works, without checking $(-x+5)^2$.

a. Test the classmate's rule using $(-x+5)^2$ — does the '+' inside this bracket predict a positive middle term as the classmate's rule suggests?

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$(-x+5)^2 = (-x)^2+2(-x)(5)+5^2 = x^2-10x+25$ — the middle term is NEGATIVE, despite the bracket containing a '+' sign. This DISPROVES the classmate's simple rule.

b. Explain why the classmate's simplified rule fails here, identifying what ACTUALLY determines the sign of the middle term (hint: think about the sign of the SECOND term specifically, not just whether there's a visible '+' or '-' symbol).

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The middle term's sign is determined by the sign of the SECOND term being squared (here, $+5$, combined with the NEGATIVE first term $-x$), not simply by which symbol appears in the bracket — the classmate's rule oversimplified by ignoring that the FIRST term itself ($-x$) can also carry a sign that affects the outcome; the correct approach is always to properly expand $(a+b)^2=a^2+2ab+b^2$ with the ACTUAL signed values of $a$ and $b$.

QUESTION 10 [7 marks] — Criterion D Hard

A square painting's canvas has side length $(x+7)$cm, and is placed inside a square frame with a UNIFORM 3cm border on all sides (so the frame's outer side length is $(x+13)$cm).

a. Write and expand expressions for the canvas area and the total (frame+canvas) area.

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Canvas: $(x+7)^2=x^2+14x+49$. Total: $(x+13)^2=x^2+26x+169$.

b. Find a simplified expression for the area of the FRAME alone (border only, excluding canvas), and evaluate it at $x=20$cm.

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Frame area $=(x^2+26x+169)-(x^2+14x+49)=12x+120$. At $x=20$: $12(20)+120=240+120=360$cm².

QUESTION 11 [7 marks] — Criterion D Hard

A city's population model predicts population $P(t)=(x+t)^2$ thousand people, where $x$ is a base constant and $t$ is years elapsed.

a. If $x=50$, expand $(50+t)^2$ to find $P(t)$ as a polynomial in $t$.

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$$P(t)=2500+100t+t^2$$

b. City planners want to know the population at $t=10$ years, and separately want to know how much of that population figure comes from the 'interaction' between the base population and time (the middle term), versus the base population itself and the pure time-squared growth. Break down $P(10)$ into its three components and their individual contributions.

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$P(10)=2500+1000+100=3600$ thousand. Component breakdown: base term ($2500$) $=69.4\%$ of total; middle/interaction term ($1000$) $=27.8\%$; time-squared term ($100$) $=2.8\%$. This shows the base population still dominates the total even after 10 years, though the interaction term contributes meaningfully, while the pure $t^2$ term remains relatively small at this stage.

Perfect square expansion