Algebraic methods

Practice

Recall it first, then use it — testing yourself builds the memory, where rereading the notes above only feels easier.

Recall and application shuffled together — you have to work out what each item wants before you can answer it, which is the part the exam actually tests. Both kinds feed the same spacing schedule, so anything you miss comes back sooner.

Say the answer out loud before flipping — that attempt is where the learning happens, even when you get it wrong.

Exam-style only. Instant feedback, shuffled each attempt, best score saved on this device.

What actually comes up

Three questions every year, each opening with two or three short simplify/solve/rearrange parts before building to a longer problem wrapped in a real scenario (a laptop's depreciating value, a company logo, a canal cross-section). The discriminant, an exponential equation solved via a substitution, and a "find the constants" problem from given roots or points all recur.

Across 3 years

The pattern — what to prepare

The 2025 paper ↗ Marking schedule ↗

Question One

Short simplify/rearrange parts, a quadratic through two given points, a rectangle inscribed in a circle, and an exponential equation.

• y = 2x² + ax + b passes through (1.5, 0) and (−1, −5)
• Rectangle inscribed in a circle of radius r, length = 2 × width

• a Simplify an expression combining powers of y (index laws). Achieved
• b Rearrange an equation of the form (2y)² = 7x − 5 to make x the subject. Achieved
• c A quadratic y = 2x² + ax + b passes through (1.5, 0) and (−1, −5). Find the values of a and b. Merit
• d A rectangle twice as long as it is wide is inscribed in a circle of radius r. Find a fully simplified expression for its area in terms of r. Excellence
• e Solve 5(5²ˣ⁺¹) − 120 = 5ˣ⁺², using the substitution n = 5ˣ. Excellence

Question Two

A perfect-square constant, an algebraic fraction to simplify, and equations to solve for unknowns a, b, k.

• 9x² − 30x + k is a perfect square
• log₅(x + 6) − log₅x = 2
• Simultaneous: y − 2x = 13 and x² + 2ky = 6k, one solution

• a Find the value of k needed to write 9x² − 30x + k as a perfect square. Achieved
• b Simplify fully a rational expression built from two quadratic factorisations. Merit
• c Solve (3x − 1)² = 5, writing the solutions in the form a ± √b, over 3, and state a and b. Merit
• d Solve log₅((x + 6)²) − log₅x = 2. Excellence
• e Given y − 2x = 13 and x² + 2ky = 6k have exactly one solution, find k and the solution. Excellence

Question Three

A log equation, a quadratic built from given roots, a log expressed in terms of p and q, a sum-of-squares identity, and a golf-shot projectile.

• A quadratic has roots 2/3 and −1/5
• log_a 3 = p, log_a 5 = q
• Ball launched 2 m above ground, max height 20 m at 110 m out; tree 7.5 m high at 200 m

• a Solve log₈x = 6. Achieved
• b A quadratic ax² + bx + c = 0 has solutions 2/3 and −1/5. Find the integer values of a, b, and c. Achieved
• c Given log_a 3 = p and log_a 5 = q, write log_a 75 in terms of p and q. Merit
• d If a and b are roots of x² − mx + n = 0, find an expression for a² + b² in terms of m and n. Excellence
• e A golf ball launched 2 m above the ground reaches a maximum height of 20 m at 110 m out. Will it clear a 7.5 m tree 200 m from the start, modelling the flight as a quadratic? Excellence

The 2024 paper ↗ Marking schedule ↗

Question One

Completing the square, a discriminant, an algebraic fraction, a cubic with known roots, and a rational identity to prove.

• 3x³ + ax² + bx + c has roots −2, 1/3, and 4

• a Write x² + 8x − 5 in the form (x + p)² − q. Achieved
• b Find the discriminant of 3x² + 2 = 5x. Achieved
• c Write 1/t + (t − 5)/t² − 1/(3t) as a single fraction. Merit
• d A cubic 3x³ + ax² + bx + c has roots −2, 1/3, and 4. Using knowledge of roots, find the values of the constants. Merit
• e Show that a given rational expression built from x^(3/2), x^(1/2), and x^(−1/2) simplifies to (x + 1)/(x(x − 1)). Excellence

Question Two

An equation solved by a substitution, a surd fraction, a discriminant condition on roots, and a company logo (circle inside a rectangle).

• x² − 2x over 4x − 1 = (k − 1)/(k + 1), roots numerically equal but opposite sign
• Logo: green rectangle of length x, with two semicircles and a circle (all same radius) cut out; green area 10 cm²

• a Solve 3x − 14 = 5/x. Achieved
• b Simplify (9x² + 30x + 25)/(5x + 3x²). Merit
• c Find k for which x² − 2x over 4x − 1 = (k−1)/(k+1) has roots numerically equal but opposite in sign. Excellence
• d(i) Find an expression in terms of x for the area of the circular "O" in a logo made of a rectangle, two semicircles, and a circle, all sharing a radius. Achieved
• d(ii) If the green background area is 10 cm², find the length of the rectangle. Merit

Question Three

A log equation, an exponential equation with fractional bases, a change-of-base relationship, a quadratic formula in terms of k, and a decibel-scale comparison.

• 9(2^(x+3)) = (1/27)^x
• log_b x = 2 and log_(3b) y = 2
• 3x² − 4kx + k² = 0
• d = 10 log₁₀(P/P₀); cooling fan 38 dB, heat pump 30 dB

• a Solve x = log₅625. Achieved
• b Solve 9(2^(x+3)) = (1/27)^x. Merit
• c If log_b x = 2 and log_(3b) y = 2, write y in terms of x. Merit
• d Using the quadratic formula, find the fully simplified solutions to 3x² − 4kx + k² = 0 in terms of k. Merit
• e A cooling fan measures 38 decibels and a heat pump 30 decibels on the scale d = 10 log₁₀(P/P₀). Show the cooling fan's sound intensity is more than six times the heat pump's. Excellence

The 2023 paper ↗ Marking schedule ↗

Question One

Two index-law simplifications, a fraction split into A + B/x form, a laptop's depreciating value, and a tangent line to a parabola.

• V = 40 + ke^(−0.5t), original price $900
• Line 2y = 2x + 29 tangent to x² − 2ky + 32k = 0

• a Simplify two expressions involving indices, leaving answers with positive indices. Achieved
• b Write (20x² − 22x + 6)/(5x² − 3x) in the form A + B/x, finding the integers A and B. Merit
• c A laptop's value follows V = 40 + ke^(−0.5t), original price $900. How long until its value drops to 50% of the original? Excellence
• d The line 2y = 2x + 29 is a tangent to x² − 2ky + 32k = 0. Find k and where the quadratic crosses the y-axis. Excellence

Question Two

A log equation, an exponential root equation, a fraction expressed as a power, a log-law evaluation, and an equation solved via substitution.

• log₂(3m + 1) = 4; 3 log_x(64) = 6
• log_b(1/b³) + log_b(√b) + 6

• a Find m if log₂(3m + 1) = 4, then solve 3·log_x(64) = 6. Achieved
• b Find an expression for p in terms of x given (5^(7x+6))/(25^(−x)) = 125^p. Merit
• c Find the value of 6 + log_b(1/b³) + log_b(√b). Excellence
• d Using an algebraic method, solve 4x − 10/(4x) = 3. Excellence

Question Three

A quadratic factorisation, an algebraic fraction to simplify, a no-real-roots condition, a rational equation for a constant k, a canal cross-section, and a candle box.

• f(x) = 2x² + 8x + p does not cross the x-axis
• (x² + 2x + k)/(x² + 7x + 10) = (x − 3)/(x + 2)
• Canal: 2.5 m wide, 3 m deep; narrowboat 2.2 m wide, needs 1 m depth
• Open box tightly fits a cylinder, matching surface areas

• a Solve 5x² + 13x = 6, then simplify (3x² − 10x + 8)/(9x² − 16). Achieved
• b Find the range of values of p for which f(x) = 2x² + 8x + p does not cross the x-axis. Merit
• c Given (x² + 2x + k)/(x² + 7x + 10) = (x − 3)/(x + 2), work out the value of k. Merit
• d A canal's cross-section is modelled as a quadratic, 2.5 m wide and 3 m deep. Will a 2.2 m-wide narrowboat needing 1 m depth float in it? Excellence
• e An open box tightly fits a cylindrical candle with matching surface areas. Write an expression for the box's height h in terms of radius r and π. Excellence

Grades above are read from each year's own Achievement/Merit/Excellence evidence statement, condensed to the actual task — the schedules also grade holistically per question on a 0–8 scale, not a fixed mark per bullet. Several exponents in the 2025 paper were reconstructed from the marking schedule's own worked solutions where the source OCR badly mangled superscripts.

Past exams & resources

⏱ Timed opens the paper and starts a countdown matching the exam length. Official NZQA PDFs. If a link 404s, that year’s paper may sit under a slightly different code: browse the NZQA subject page.