Algebra

Worked examples (12)extended set

• 1Rearranging a linear formula
• 2Subject inside a squared term
• 3Subject under a square root
• 4Subject in a denominator
• 5Rearranging with two terms
• 6Subject appearing twice
• 7Sector perimeter: build the formula, then rearrange for the angle
• 8Cone: rearranging a formula with a square root
• 9Window frame: the subject appears twice, with π
• 10Isolating a squared term with a negative coefficient
• 11Rearranging when the subject has a negative sign
• 12Subject in a fraction and a bracket: collect and factorise

Worked examples (11)extended set

• 1Substituting into a linear formula
• 2Substituting into a formula containing a square
• 3Substituting into an algebraic fraction
• 4Negative values with an even power
• 5Inverse square relationship
• 6Substituting then solving for an unknown
• 7Reading values from a diagram — watch the units
• 8Frustum volume from diameters, then working backwards
• 9Arch formula: show that, then a quadratic in context
• 10Negative value, then a square root
• 11Substitute forwards, then work backwards to a square

Worked examples (13)extended set

• 1Collecting like terms
• 2Expanding and subtracting a bracket
• 3Expanding double brackets
• 4Index laws with algebraic terms
• 5Dividing algebraic terms
• 6Difference of two expanded squares
• 7Missing sides and the perimeter of an L-shape
• 8Area of a rectangle with a corner removed
• 9Frame split into trapezia — two routes to one expression
• 10Expanding three brackets, one of them squared
• 11Expanding a bracket with powers, then collecting
• 12A fractional power of a term with a coefficient
• 13Products and sums: which terms can combine?

Worked examples (12)extended set

• 1Taking out a common factor
• 2Difference of two squares
• 3Quadratic with leading coefficient 1
• 4Quadratic with a leading coefficient
• 5Factorising fully — two techniques combined
• 6Factorising by grouping
• 7Completing an area model to factorise
• 8Running track: factorising a formula containing π
• 9Difference of two squares inside a right-angled triangle
• 10Grouping when the second factor is 1
• 11Exact x-intercepts of a cubic by factorising
• 12Using a factorised form to find integer solutions

Worked examples (13)extended set

• 1Unknown on both sides
• 2Equation with algebraic fractions
• 3Inequality with a negative coefficient
• 4Double inequality and integer solutions
• 5Forming and solving an equation
• 6Inequality from a context, with a bound
• 7Forming an equation from the angles of a quadrilateral
• 8Reading a number line and combining two inequalities
• 9Triangle with algebraic sides: finding the range of x
• 10Equation with a bracket and a fractional answer
• 11Forming an equation from ticket prices
• 12Finding a missing frequency from the mean
• 13Area of two trapezia in reverse

Worked examples (12)extended set

• 1Solving by factorising
• 2Solving by factorising with a leading coefficient
• 3Using the quadratic formula
• 4Completing the square
• 5Forming and solving a quadratic
• 6Quadratic inequality
• 7Pythagoras leading to a quadratic
• 8Solving quadratic inequalities from a graph
• 9Frame with unequal borders: spot the error, then solve
• 10A ferry slowed by the wind
• 11Using symmetry to write down the other root
• 12Exact answers from the formula, without a calculator

Worked examples (11)extended set

• 1Elimination by adding
• 2Elimination needing both equations multiplied
• 3Solving by substitution
• 4Forming simultaneous equations from a context
• 5One linear, one quadratic
• 6Simultaneous equations with fractions
• 7Forming simultaneous equations from a rectangle
• 8Graphical solution: estimate, then solve exactly
• 9Where a line cuts a curve: coordinates and length
• 10Line and curve that need the formula
• 11Angles in a parallelogram

Worked examples (13)extended set

• 1nth term from a growing pattern
• 2nth term of a fraction sequence
• 3Quadratic sequence from second differences
• 4Geometric sequence: nth term and a later term
• 5Solving for the position of a term
• 6Proving a bound on a fraction sequence
• 7Linear nth term and a far-off term
• 8Tile patterns: two sequences from one diagram
• 9Staircase patterns: a quadratic and a linear sequence
• 10Shaded triangles: a geometric sequence and a threshold
• 11Cubic, fraction and geometric sequences
• 12A decreasing linear sequence
• 13A formula for the sum of the first n terms

Worked examples (11)extended set

• 1Direct variation with a cube
• 2Inverse variation with a square root
• 3Identifying the type of variation from a table
• 4Directly with one variable, inversely with another
• 5Percentage change when y varies as x squared
• 6Percentage change with an inverse cube
• 7Finding the constant of proportionality
• 8Matching variation statements to sketch graphs
• 9Braking distance: finding the formula from a graph
• 10Inverse square law with a lamp post
• 11Inverse proportion to the root of a bracket

Worked examples (13)extended set

• 1Simplifying a simple algebraic fraction
• 2Simplifying by factorising first
• 3Adding algebraic fractions
• 4Subtracting algebraic fractions
• 5Dividing algebraic fractions
• 6Solving an equation containing algebraic fractions
• 7Perimeter of a triangle with fractional sides
• 8Two-stage journey: forming an equation with fractions
• 9Resistors in a circuit: combine, factorise, solve
• 10Three fractions with numerical denominators
• 11Factorising a quadratic and a four-term expression, then cancelling
• 12Common factor on top, difference of two squares below — with a sign change
• 13Multiplying fractions and simplifying fully

Worked examples (12)extended set

• 1Stationary points and their nature
• 2Equation of a tangent
• 3Unknown power and coefficient from the derivative
• 4Where a curve is increasing or decreasing
• 5Optimisation: the largest open box from a sheet of card
• 6Finding unknown coefficients from a turning point
• 7Differentiate and evaluate at a point
• 8Gradient from a drawn tangent, then by differentiation
• 9Two tangents meeting below a curve
• 10Largest rectangle under a curve
• 11Turning points, then how many solutions
• 12Which turning point is the maximum?

Worked examples (12)extended set

• 1Converting km/h to m/s before finding acceleration
• 2Total distance from a trapezium-shaped speed-time graph
• 3One vehicle catching another
• 4Average speed on a return journey
• 5Finding an unknown time from the total distance
• 6Estimating distance under a curved speed-time graph
• 7Reading a distance-time graph
• 8Speed-time graph that does not start from rest
• 9Speed at an instant from a tangent on a distance-time graph
• 10When does the accelerating car catch up?
• 11A speed-time graph with a curved section
• 12A lorry joining a motorway: unknown starting speed