Quadratics, Simultaneous Equations & Inequalities

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Corbett MathsSolving quadratics — factorising, formula, completing the square

Corbett MathsSimultaneous equations including quadratic

GCSE Maths TutorCompleting the Square & Turning Points (Higher Only) | GCSE Maths Tutor

GCSE Maths TutorSimultaneous Equations (Higher & Foundation) | GCSE Maths Tutor

Key facts & methods

Solving quadratic equations

Factorisation: factorise → set each bracket to 0. Works when the quadratic factorises neatly.
Quadratic formula: x = (−b ± √(b²−4ac)) / 2a. Works always. Memorise this.
Completing the square: write in form (x+p)²=q, then x = −p ± √q.
Discriminant b²−4ac: if >0 → two solutions. If =0 → one solution. If <0 → no real solutions.

Simultaneous equations

Linear simultaneous equations: elimination (multiply to match a coefficient, then add/subtract) or substitution.
Elimination: if coefficients of y are 2y and 3y → multiply first equation ×3 and second ×2, then subtract.
One linear, one quadratic: rearrange the linear for one variable, substitute into the quadratic → solve quadratic.
The solutions give coordinates of intersection of a line and a curve.

Inequalities

Linear inequalities: solve like equations, but FLIP the inequality when multiplying/dividing by a negative.
E.g. −2x < 6 → x > −3 (flipped because divided by −2).
Represent on number line: open circle (○) for strict < or >. Closed circle (●) for ≤ or ≥.
Quadratic inequalities: solve the equation → sketch the curve → decide which region satisfies the inequality.
E.g. x²−5x+6 < 0 → (x−2)(x−3) < 0 → 2 < x < 3 (between the roots, parabola is below x-axis).

Exam questions — 4 questions · 10 marks · Edexcel 4MA1 style

4 marksSolve quadratic equation4MA1 style ✏️ No calculator

Solve x² + 3x − 10 = 0 by (a) factorisation. (b) Also solve 2x² − 5x − 3 = 0 using the quadratic formula. Give answers to 2 d.p. (4 marks)

Hint: (a) Find two numbers that multiply to −10 and add to +3. (b) Identify a, b, c then substitute carefully into x = (−b ± √(b²−4ac)) / 2a.

+40 XP

4 marksSimultaneous equations (linear)4MA1 Set 1 style ✏️ No calculator

Solve the simultaneous equations: 3x + 2y = 12 and 5x − y = 13. (4 marks)

Hint: Try substitution: rearrange the simpler equation for y (or x), substitute into the other. Or use elimination by multiplying to match coefficients.

+40 XP

4 marksSimultaneous — linear and quadraticGrade 7 Booklet style / 4MA1 style ✏️ No calculator

Solve the simultaneous equations: y = 2x + 1 and y = x² − 2x + 3. (4 marks)

Hint: Substitute the linear expression for y into the quadratic. Rearrange to get 0 on one side, then solve using the quadratic formula.

+40 XP

3 marksQuadratic inequalityGrade 7 Booklet style ✏️ No calculator

Solve the quadratic inequality x² − 5x + 4 ≤ 0. (3 marks)

Hint: First solve the equation (=0). Then sketch the parabola. Since the coefficient of x² is positive, the parabola opens upward — it is below the x-axis BETWEEN the roots.

+30 XP

1 markDiscriminant4MA1 style

The equation 2x² + kx + 8 = 0 has exactly one solution. What is a possible value of k?

A 4

B 8

C 10

D 0

Quadratics, Simultaneous Equations & Inequalities

Solving quadratic equations

Simultaneous equations

Inequalities

Module complete! 🎉