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Find the complete real solution set of a rational quadratic inequality · Check your progress
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Find every real solution of a rational line–quadratic system · Check your progress
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Derive complete parameter sets for the real roots of a quadratic · Check your progress
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Derive complete parameter sets for line–quadratic intersections · Check your progress
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Derive complete parameter sets for a quadratic to be always positive or negative · Check your progress
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Produce the complete polynomial quotient and remainder · Check your progress
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Fully factor a cubic and give every distinct real root · Check your progress
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Decompose partial fractions with two distinct linear factors · Check your progress
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Decompose partial fractions with a repeated linear factor · Check your progress
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Decompose partial fractions with a linear and irreducible quadratic factor · Check your progress
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Simplify surd sums, differences and products exactly · Check your progress
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Rationalise surd denominators exactly · Check your progress
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Solve a linear equation with surd coefficients exactly · Check your progress
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Solve a radical equation completely and reject every extraneous candidate · Check your progress
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Complete the square to find the greatest or least value of a quadratic · Check your progress
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Find the complete real solution set of a rational quadratic inequality · Check again
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Find every real solution of a rational line–quadratic system · Check again
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Derive complete parameter sets for the real roots of a quadratic · Check again
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Derive complete parameter sets for line–quadratic intersections · Check again
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Derive complete parameter sets for a quadratic to be always positive or negative · Check again
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Produce the complete polynomial quotient and remainder · Check again
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Fully factor a cubic and give every distinct real root · Check again
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Decompose partial fractions with two distinct linear factors · Check again
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Decompose partial fractions with a repeated linear factor · Check again
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Decompose partial fractions with a linear and irreducible quadratic factor · Check again
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Simplify surd sums, differences and products exactly · Check again
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Rationalise surd denominators exactly · Check again
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Solve a linear equation with surd coefficients exactly · Check again
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Solve a radical equation completely and reject every extraneous candidate · Check again
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Complete the square to find the greatest or least value of a quadratic · Check again
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Find the complete real solution set of a rational quadratic inequality · Retention check
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Find every real solution of a rational line–quadratic system · Retention check
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Derive complete parameter sets for the real roots of a quadratic · Retention check
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Derive complete parameter sets for line–quadratic intersections · Retention check
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Derive complete parameter sets for a quadratic to be always positive or negative · Retention check
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Produce the complete polynomial quotient and remainder · Retention check
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Fully factor a cubic and give every distinct real root · Retention check
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Decompose partial fractions with two distinct linear factors · Retention check
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Decompose partial fractions with a repeated linear factor · Retention check
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Decompose partial fractions with a linear and irreducible quadratic factor · Retention check
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Simplify surd sums, differences and products exactly · Retention check
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Rationalise surd denominators exactly · Retention check
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Solve a linear equation with surd coefficients exactly · Retention check
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Solve a radical equation completely and reject every extraneous candidate · Retention check
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Complete the square to find the greatest or least value of a quadratic · Retention check
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Practise: Find the complete real solution set of a rational quadratic inequality
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Practise: Find every real solution of a rational line–quadratic system
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Practise: Derive complete parameter sets for the real roots of a quadratic
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Practise: Derive complete parameter sets for line–quadratic intersections
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Practise: Derive complete parameter sets for a quadratic to be always positive or negative
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Practise: Produce the complete polynomial quotient and remainder
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Practise: Fully factor a cubic and give every distinct real root
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Practise: Decompose partial fractions with two distinct linear factors
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Practise: Decompose partial fractions with a repeated linear factor
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Practise: Decompose partial fractions with a linear and irreducible quadratic factor
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Practise: Simplify surd sums, differences and products exactly
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Practise: Rationalise surd denominators exactly
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Practise: Solve a linear equation with surd coefficients exactly
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Practise: Solve a radical equation completely and reject every extraneous candidate
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Practise: Complete the square to find the greatest or least value of a quadratic
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Revision notes
Connect quadratic models, discriminants and inequalities with surds, polynomial factors and partial fractions in G3 Additional Mathematics.
Key ideas
- Identities and general quadratic factors
Derive useful identities by distribution and verify every factorisation by expansion.
- Zero products and quadratic equations
Set a factored quadratic equal to zero and retain every factor alternative.
- Completing the square and fractional quadratics
Choose an exact quadratic method and distinguish solving from rewriting a function.
- Surds and extraneous roots
Calculate with exact surds, rationalise denominators, solve with surd coefficients and check candidates after each squaring step.
- Quadratic extrema and sign
Connect completed squares and discriminants to strict sign conditions, model restrictions and feasible quadratic extrema.
- Discriminants, intersections and inequalities
Connect substitution, root counts and tangency to sign intervals, number-line solutions and parameter cases that change degree.
- Polynomial division and the factor theorem
Carry out aligned division, use factor/remainder conditions and verify complete cubic factors and distinct real solutions.
- Partial fraction structure
Separate polynomial parts, derive coefficients for each scoped factor pattern and verify identities with original exclusions.
From the G3 Additional Mathematics 2027 syllabus
Show all 21 syllabus statements
- Finding the maximum or minimum value of a quadratic function using the method of completing the square
- Conditions for y = ax^2 + bx + c to be always positive (or always negative)
- Using quadratic functions as models
- Conditions for a quadratic equation to have:
- Two real roots
- Two equal roots
- no real roots and related conditions for a given line to:
- Intersect a given curve
- be a tangent to a given curve
- Not intersect a given curve
- Solving simultaneous equations in two variables by substitution, with one of the equations being linear equation
- Solving quadratic inequalities, and representing the solution on the number line
- Four operations on surds, including rationalising the denominator
- Solving equations involving surds
- Multiplication and division of polynomials
- Use of remainder and factor theorems, including factorising polynomials and solving cubic equations
- Use of: a^3 + b^3 = (a + b)(a^2 − ab + b^2 ) a^3 − b^3 = (a − b)(a^2 + ab + b^2 )
- Partial fractions with cases where the denominator is no more complicated than:
- (ax + b)(cx + d)
- (ax + b)(cx + d)²
- (ax + b)(x² + c² )