LAWS OF ALGEBRA

QUESTION 1 [4 marks] — Criterion A Medium

Expand each expression:

a. $(x+8)(x-8)$

[1]

Show Solution

$$x^2-64$$

b. $(2x+3)(2x-3)$

[2]

Show Solution

$$4x^2-9$$

c. $(x-9)(x+9)$

[1]

Show Solution

$$x^2-81$$

QUESTION 2 [4 marks] — Criterion B Medium

Investigate why the 'middle terms' always cancel out in a difference-of-squares expansion, using $(x+a)(x-a)$.

a. Fully expand $(x+a)(x-a)$ using FOIL, showing all four terms before simplifying.

[2]

Show Solution

$(x+a)(x-a) = x^2 -ax+ax-a^2$ (First: $x^2$, Outer: $-ax$, Inner: $ax$, Last: $-a^2$).

b. Explain why the middle two terms always cancel, for ANY value of $a$.

[2]

Show Solution

The Outer term is $-ax$ and the Inner term is $+ax$ — these are always exact opposites of each other (same size, opposite sign) regardless of what $a$ is, so they always sum to zero and cancel out, leaving just $x^2-a^2$.

QUESTION 3 [5 marks] — Criterion C Medium

A student is confused why $(x+6)(x-6)$ expands to $x^2-36$ (no middle term), while $(x+6)(x+6)$ expands to $x^2+12x+36$ (WITH a middle term).

a. Expand both expressions fully to confirm these results.

[3]

Show Solution

$(x+6)(x-6)=x^2-6x+6x-36=x^2-36$ (middle terms cancel). $(x+6)(x+6)=x^2+6x+6x+36=x^2+12x+36$ (middle terms ADD instead).

b. Explain the key difference between the two expressions that causes this different behaviour.

[2]

Show Solution

In $(x+6)(x-6)$, the second bracket has a MINUS sign, making the Outer and Inner terms opposite in sign (so they cancel). In $(x+6)(x+6)$, both brackets have a PLUS sign, so the Outer and Inner terms have the SAME sign (so they add together instead of cancelling).

QUESTION 4 [5 marks] — Criterion D Medium

A mental-maths trick: to calculate $47\times53$ quickly, write it as $(50-3)(50+3)$.

a. Use the difference-of-squares pattern to evaluate $(50-3)(50+3)$ without a calculator.

[2]

Show Solution

$$(50-3)(50+3)=50^2-3^2=2500-9=2491$$

b. Verify this matches $47\times53$ by direct multiplication, and explain why this 'trick' works for any two numbers that are equally spaced above and below a round number.

[3]

Show Solution

$47\times53=2491$ ? (matches). This works because any such pair can be written as $(n-a)(n+a)=n^2-a^2$, which is often much easier to calculate than direct multiplication when $n$ is a round number like 50.

QUESTION 5 [6 marks] — Criterion B Hard

Investigate a mental-maths trick for multiplying numbers close to a round number, using $97\times103$.

a. Write $97\times103$ in the form $(100-a)(100+a)$ for an appropriate value of $a$, then apply the difference-of-squares pattern to evaluate it without direct multiplication.

[3]

Show Solution

$97\times103=(100-3)(100+3)=100^2-3^2=10000-9=9991$.

b. Verify this matches direct multiplication, then explain why this 'trick' only works efficiently when the two numbers are EQUALLY spaced above and below a convenient round number.

[3]

Show Solution

Direct check: $97\times103=9991$ ?. The trick relies specifically on the pattern $(n-a)(n+a)=n^2-a^2$, which requires the two numbers to be exactly $a$ below and $a$ above some central value $n$ — if the numbers aren't symmetric around a round number this way, they can't be written in this convenient $(n-a)(n+a)$ form, and the shortcut doesn't apply directly.

QUESTION 6 [3 marks] — Criterion C Medium

A student expands $(5x-2)(5x+2)$ as $5x^2-4$ (forgetting to square the coefficient 5 as well as the $x$).

a. Explain the error precisely, and give the correct expansion.

[3]

Show Solution

The student only squared the VARIABLE part ($x\to x^2$) but forgot the COEFFICIENT must also be squared: $(5x)^2=25x^2$, not $5x^2$. Correct: $(5x-2)(5x+2)=(5x)^2-2^2=25x^2-4$.

QUESTION 7 [7 marks] — Criterion D Hard

A square garden plot has side length $(x+20)$m. A smaller square garden plot has side length $(x-20)$m, made from the same available materials budget scaled differently.

a. Write and expand an expression for the DIFFERENCE in area between the two plots (larger minus smaller).

[3]

Show Solution

$$(x+20)^2-(x-20)^2$$ Using the earlier pattern $(x+a)^2-(x-a)^2=4ax$ with $a=20$: $$=80x$$

b. Alternatively, this difference could be found by factoring $(x+20)^2-(x-20)^2$ as a difference of squares FIRST: $[(x+20)+(x-20)][(x+20)-(x-20)]$. Simplify this factored form, and confirm it matches your answer to part (a).

[4]

Show Solution

$[(x+20)+(x-20)][(x+20)-(x-20)] = [2x][40] = 80x$ — matches part (a) exactly, confirming both approaches (expanding perfect squares directly, or factoring as a difference of squares first) give the same correct result.

QUESTION 8 [7 marks] — Criterion D Hard

A manufacturer produces circular metal discs. The area difference between a disc of radius $(r+2)$cm and a smaller disc of radius $(r-2)$cm (using $\text{Area}=\pi r^2$, and treating the DIFFERENCE OF SQUARES pattern on the radii) needs to be calculated for material cost estimates.

a. Write an expression for the area difference, $\pi(r+2)^2 - \pi(r-2)^2$, and simplify using the difference-of-squares / expansion pattern.

[4]

Show Solution

$$\pi[(r+2)^2-(r-2)^2] = \pi[4(2)r] = 8\pi r$$ (using the pattern $(x+a)^2-(x-a)^2=4ax$ with $a=2$).

b. If $r=15$cm, find the exact area difference (in terms of $\pi$) and as a decimal correct to 1 decimal place.

[3]

Show Solution

$8\pi(15)=120\pi \approx 376.99 \approx 377.0 \text{ cm}^2$.

QUESTION 9 [6 marks] — Criterion D Medium

A solar panel installation compares two square panel arrangements: a large array with side length $(x+50)$cm and a small array with side length $(x-50)$cm, where $x$ represents a base module size.

a. Write and simplify an expression for the difference in area between the two arrangements.

[3]

Show Solution

$(x+50)^2-(x-50)^2 = 4(50)x = 200x$ (using the established pattern).

b. The installer needs the area difference to be AT LEAST 8000 cm² to justify the cost of the larger array. Find the minimum value of $x$ that satisfies this requirement.

[3]

Show Solution

$200x\ge8000 \Rightarrow x\ge40$cm.

Difference of two squares