About this topic
Before you begin
Connect quadratic models, discriminants and inequalities with surds, polynomial factors and partial fractions in G3 Additional Mathematics.
Learning goals
- Manipulate surds and rationalise denominators
- Simplify surd sums, differences and products exactly
- Rationalise surd denominators exactly
- Solve a linear equation with surd coefficients exactly
- Solve a radical equation completely and reject every extraneous candidate
- Derive complete parameter sets for a quadratic to be always positive or negative
- Complete the square to find the greatest or least value of a quadratic
- Formulate, use and interpret a mathematical model
- Use quadratic structure and discriminants to reason about roots
- Interpret and draw graphs of functions
- Derive complete parameter sets for the real roots of a quadratic
- Derive complete parameter sets for line–quadratic intersections
- Find the complete real solution set of a rational quadratic inequality
- Find every real solution of a rational line–quadratic system
- Solve simultaneous equations by justified methods
- Solve inequalities and represent their solution sets
- Produce the complete polynomial quotient and remainder
- Fully factor a cubic and give every distinct real root
- Divide and factor polynomials using remainder and factor theorems
- Decompose partial fractions with two distinct linear factors
- Decompose partial fractions with a repeated linear factor
- Decompose partial fractions with a linear and irreducible quadratic factor
- Decompose appropriate rational expressions into partial fractions
Syllabus statements covered
- 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 )
- 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 )
Choose a lesson
Work through them in order.
- Not startedIdentities and general quadratic factorsNextDerive useful identities by distribution and verify every factorisation by expansion.
- Not startedZero products and quadratic equationsNextSet a factored quadratic equal to zero and retain every factor alternative.
- Not startedCompleting the square and fractional quadraticsNextChoose an exact quadratic method and distinguish solving from rewriting a function.
- Not startedSurds and extraneous rootsNextCalculate with exact surds, rationalise denominators, solve with surd coefficients and check candidates after each squaring step.
- Not startedQuadratic extrema and signNextConnect completed squares and discriminants to strict sign conditions, model restrictions and feasible quadratic extrema.
- Not startedDiscriminants, intersections and inequalitiesNextConnect substitution, root counts and tangency to sign intervals, number-line solutions and parameter cases that change degree.
- Not startedPolynomial division and the factor theoremNextCarry out aligned division, use factor/remainder conditions and verify complete cubic factors and distinct real solutions.
- Not startedPartial fraction structureNextSeparate polynomial parts, derive coefficients for each scoped factor pattern and verify identities with original exclusions.