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Quadratics, Simultaneous Equations and Inequalities

Completing the square finds the turning point of a quadratic; the discriminant b2−4acb^2-4ac tells you how many roots it has. Solve a line and a curve together by substitution, and solve quadratic inequalities by sketching the parabola and reading off where it is above or below the axis.

Quadratic Equations

Rearrange to ax2+bx+c=0ax^2+bx+c=0. If it factorises, each bracket gives a solution. Otherwise use the quadratic formula, or complete the square, which also gives the turning point of the graph. In context, reject solutions that make no sense.

Solving Quadratic Equations

Three ways to solve ax2+bx+c=0ax^2+bx+c=0 (factorising, completing the square and the formula), choosing the quickest, giving exact answers, and spotting quadratics in disguise such as x4−5x2+4=0x^4-5x^2+4=0.

Linear and Quadratic Inequalities

Solve linear inequalities, then quadratic ones using critical values and a sketch, write the answer correctly (and, or, set notation), and combine two conditions.

Populations and Sampling

A population is all the organisms of one species in an area; a community is all the populations of different species; an ecosystem is a community and its physical environment. Population size depends on biotic factors (food, predators, disease, competition) and abiotic factors (temperature, light, water). Populations often follow a sigmoid curve: lag, exponential (log) growth, stationary (births = deaths) and sometimes death phase. Population size is estimated with quadrats: count organisms in random quadrats, find the mean, and scale up to the total area.

Solving Trigonometric Equations

How to find every solution of a trig equation in an interval: principal value, symmetry, and period. Then equations with 2x2x or x+30∘x+30^\circ inside, quadratics in sin⁡x\sin x or cos⁡x\cos x, and equations that need an identity first.

Tangents, Chords and Intersections

The two right angles that unlock circle questions (tangent and radius, centre and chord), and how substituting a line into a circle or curve turns geometry into a quadratic whose discriminant tells you how they meet.

Exponential Equations and Inequalities

Bringing an unknown power down with logarithms: solving ax=ba^x=b, equations with xx in two powers, disguised quadratics, and inequalities like "the smallest nn such that 0.8n<0.10.8^n<0.1".

Polynomial Equations and Square Roots

Every quadratic now has two roots. For polynomials with real coefficients, complex roots come in conjugate pairs, which lets you find all the roots from one. Square roots of a+bia+bi come from equating real and imaginary parts.

Travel Graphs and Curves

On a distance–time graph the gradient is the speed. On a speed–time graph the gradient is the acceleration and the area underneath is the distance. Learn the shapes of quadratic, cubic, reciprocal and exponential graphs so you can recognise them at a glance.

Syllabus topics

TopicBoardPaperLesson
Linear and quadratic inequalitiesCambridge 9709Pure Mathematics 1Available
Solving quadratic equationsCambridge 9709Pure Mathematics 1Available
Quadratic equationsCambridge 4024CalculatorAvailable
Quadratic equationsCambridge 4024Non-calculatorAvailable
Quadratics, simultaneous equations and inequalitiesEdexcel YMA01Pure Mathematics 1Available
Linear and quadratic inequalitiesEdexcel 9MA0Pure Mathematics 1Available
Solving quadratic equationsEdexcel 9MA0Pure Mathematics 1Available
Linear and quadratic inequalitiesEdexcel 9MA0Pure Mathematics 2Available
Solving quadratic equationsEdexcel 9MA0Pure Mathematics 2Available
Quadratic equationsEdexcel 4MA1Higher tierAvailable
Quadratic equationsEdexcel 4MA1Foundation tierAvailable
Solving quadratic equationsAQA 7357Pure and MechanicsAvailable
Solving quadratic equationsAQA 7357Pure and StatisticsAvailable
Linear and quadratic inequalitiesAQA 7357PureAvailable
Solving quadratic equationsAQA 7357PureAvailable
Linear and quadratic inequalitiesAQA 7357Pure and MechanicsAvailable
Linear and quadratic inequalitiesAQA 7357Pure and StatisticsAvailable
Quadratic equationsAQA 8300Foundation tierAvailable
Quadratic equationsAQA 8300Higher tierAvailable
QuadraticsCambridge 9709Pure Mathematics 1Planned

Questions

IDQuestionTopicMarksDifficulty
SA-9709-P1-QD-008What is the minimum value of (x-3)^2+7?…Quadratics: completing the square and the vertex
Vertex, line of symmetry and least or greatest value		1L1 Foundation
SA-9709-P1-QD3-Q02The curve y=ax^2+bx+c touches the x-axis without crossing it. Then…Quadratics: the discriminant
Using b² − 4ac to find the number of roots		1L1 Foundation
SA-9709-P1-QD2-G01Solve x^2-7x+10=0.…Quadratics: solving quadratic equations
Solving by factorising		2L1 Foundation
SA-9709-P1-CG-Q03A line meets a curve, giving a quadratic equation. The line is a tangent when the discriminant is…Coordinate geometry: tangents, chords and intersections
Where a line meets a curve or circle		1L1 Foundation
SA-9709-P1-QD-Q01x^2+10x is equal to…Quadratics: completing the square and the vertex
Completing the square, a = 1		1L1 Foundation
SA-9709-P1-QD-Q02The vertex of y=(x+2)^2-7 is…Quadratics: completing the square and the vertex
Vertex, line of symmetry and least or greatest value		1L1 Foundation
SA-9709-P1-QD-017For the equation 2x^2-3x+5=0, find the value of the discriminant and state the number of real roots.…Quadratics: the discriminant
Using b² − 4ac to find the number of roots		2L1 Foundation
SA-9709-P1-QD-001Express x^2+6x+11 in the form (x+a)^2+b, where a and b are constants.…Quadratics: completing the square and the vertex
Completing the square, a = 1		2L1 Foundation
SA-9709-P1-QD-G01x^2-4x+9=(x+p)^2+q. Work through it in three steps.…Quadratics: completing the square and the vertex
Completing the square, a = 1		3L1 Foundation
SA-9709-P1-QD-002Express x^2-10x+4 in the form (x+a)^2+b.…Quadratics: completing the square and the vertex
Completing the square, a = 1		2L1 Foundation
SA-9709-P1-QD-010Expand and simplify (x-3)^2-4. (Algebraic answers are checked for equivalence, so any correct form is accepted.)…Quadratics: completing the square and the vertex
Completing the square, a = 1		1L1 Foundation
SA-9709-P1-QD-Q05A quadratic equation has discriminant 0. How many real roots does it have?…Quadratics: the discriminant
Using b² − 4ac to find the number of roots		1L1 Foundation
SA-9709-P1-QD-Q06The solution of (x-1)(x-5)<0 is…Quadratics: inequalities
Solving quadratic inequalities		1L1 Foundation
SA-9709-P1-QD2-Q01A student solves x^2=7x by dividing both sides by x. Which solution do they lose?…Quadratics: solving quadratic equations
Solving by factorising		1L1 Foundation
SA-9709-P3-CX4-009The quadratic with roots 14i is…Complex numbers: polynomial equations and square roots
Complex roots of polynomials and conjugate pairs		1L1 Foundation