Quadratics and algebra
Connect quadratic models, discriminants and inequalities with surds, polynomial factors and partial fractions.
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Before you begin
Connect quadratic models, discriminants and inequalities with surds, polynomial factors and partial fractions.
Learning goals
- Manipulate surds and rationalise denominators
- Use quadratic structure and discriminants to reason about roots
- Interpret and draw graphs of functions
- Formulate, use and interpret a mathematical model
- Solve simultaneous equations by justified methods
- Solve inequalities and represent their solution sets
- Divide and factor polynomials using remainder and factor theorems
- 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 )
Lessons
Work through them in order.
About lesson progress
Lesson marks record what you have studied. Checks show understanding separately. Guest marks come from this device.
- Identities and general quadratic factorsDerive useful identities by distribution and verify every factorisation by expansion.
- Zero products and quadratic equationsSet a factored quadratic equal to zero and retain every factor alternative.
- Completing the square and fractional quadraticsChoose an exact quadratic method and distinguish solving from rewriting a function.
- Surds and extraneous rootsCalculate with exact surds, track conjugate and cancellation restrictions, and check candidates after each squaring step.
- Quadratic extrema and signConnect completed squares and discriminants to strict sign conditions, model restrictions and feasible quadratic extrema.
- Discriminants, intersections and inequalitiesConnect substitution, root counts and tangency to sign intervals, number-line solutions and parameter cases that change degree.
- Polynomial division and the factor theoremCarry out aligned division, use factor/remainder conditions and verify complete cubic factors and distinct real solutions.
- Partial fraction structureSeparate polynomial parts, derive coefficients for each scoped factor pattern and verify identities with original exclusions.