Almost every algebra question in O-Level Paper 1 and Paper 2 rests on solving an equation cleanly. That includes simultaneous equations, quadratics, graph work and coordinate geometry, and later the whole of A-Math. An inequality question also appears on almost every paper, either as a direct “solve” item or inside a discriminant or graph question.
The notes start with linear and fractional equations and move on to simultaneous equations and four methods for solving quadratics. They then cover forming equations from word problems, the discriminant and line–curve intersection, and finally linear and quadratic inequalities, ending with a short preview of JC-level extensions.
What these notes cover
- •Linear and fractional equations
- •Simultaneous linear equations: substitution, elimination and graphical
- •No solution or infinitely many solutions
- •One linear and one non-linear equation
- •Quadratic equations: four ways to solve
- •Fractional equations that reduce to quadratics
- •Formulating equations from word problems
- •The discriminant b² − 4ac
- •Line–curve intersection and range of k
- •Linear inequalities and number lines
- •Quadratic inequalities
- •JC preview: fractional and modulus inequalities
Key definitions
- Linear equation
- An equation in which the unknown appears only to the power 1: no x², no √x and no x in a denominator. A linear equation in one variable has exactly one solution.
- Solution of simultaneous equations
- A pair of values (x, y) that satisfies both equations at the same time. On a graph, this is the point where the two lines intersect.
- Quadratic equation
- Any equation that can be written as ax² + bx + c = 0 with a ≠ 0. It has at most two real roots.
- Discriminant
- The expression b² − 4ac under the square root in the quadratic formula. Its sign shows which case applies: two real distinct roots, two real equal roots, or no real roots.
- Zero-product principle
- If A × B = 0, then A = 0 or B = 0. This is why an equation must be rearranged so that one side is exactly 0 before factorising.
Key formulas
Quadratic formula
For ax² + bx + c = 0. Rearrange to = 0 first and put negative values of b and c in brackets. It works on every quadratic. Give non-exact answers to 3 s.f. unless the question states otherwise.
Completing the square
Applies when the leading coefficient is 1. The curve y = x² + px + q has minimum point (−p ÷ 2, q − p² ÷ 4) and line of symmetry x = −p ÷ 2.
Discriminant
Discriminant = b² − 4ac
> 0: two real distinct roots. = 0: two real equal roots. < 0: no real roots. ≥ 0: real roots, either distinct or equal. Apply it to the quadratic obtained after substituting a line into a curve: > 0 means the line cuts the curve twice, = 0 means a tangent, < 0 means they do not meet.
Word-problem expressions
time = distance ÷ speed; amount for one = total amount ÷ number of items
These two relationships build almost every expression in word problems. Keep units consistent: 45 minutes enters the equation as ¾ hour, and $20 as 2000 cents.
The flip rule
If a < b and k < 0, then ka > kb
Multiplying or dividing both sides of an inequality by a negative number reverses the sign. Adding, subtracting, or multiplying or dividing by a positive number leaves the sign unchanged.
Quadratic inequality read-off
(x − α)(x − β) < 0 gives α < x < β; (x − α)(x − β) > 0 gives x < α or x > β
For a U-shaped parabola (positive x² coefficient) with roots α < β. “Less than zero” means between the roots, and “greater than zero” means outside them.
Worked example
Find the range of values of k for which the line y = 3x + 7 meets the curve y = x² + (k + 4)x + 3k + 5 at two distinct points.
- 1At an intersection point the y-values agree, so x² + (k + 4)x + 3k + 5 = 3x + 7.
- 2Bring every term to the left: x² + (k + 4 − 3)x + (3k + 5 − 7) = 0, which simplifies to x² + (k + 1)x + (3k − 2) = 0.
- 3Read off the coefficients of this quadratic: a = 1, b = k + 1, c = 3k − 2.
- 4Two distinct intersection points means b² − 4ac > 0, so (k + 1)² − 4(1)(3k − 2) > 0.
- 5Expand carefully: k² + 2k + 1 − 12k + 8 > 0, which gives k² − 10k + 9 > 0.
- 6Factorise: (k − 1)(k − 9) > 0.
- 7The parabola in k opens upwards and crosses the axis at k = 1 and k = 9. It lies above the axis outside the roots, so k < 1 or k > 9.
Watch out: Join the two pieces with “or”, never “and”, and never write 9 < k < 1. If the question only asked for the line to meet the curve, with a tangent allowed, the condition would be b² − 4ac ≥ 0 and the answer would be k ≤ 1 or k ≥ 9.
Common mistakes
Multiplying only some terms by the LCM
Wrong: From x ÷ 3 + (x − 2) ÷ 4 = 3, writing 4x + 3(x − 2) = 3.
Right: Multiply every term by 12, including the right-hand side: 4x + 3(x − 2) = 36, so 7x = 42 and x = 6.
Dividing by the unknown
Wrong: Dividing 5m² + 30m = 0 by m to get 5m + 30 = 0, which leaves only m = −6.
Right: Factor out 5m instead: 5m(m + 6) = 0, so m = 0 or m = −6.
Sign slip when expanding −4ac
Wrong: −4(3k − 2) = −12k − 8
Right: −4(3k − 2) = −12k + 8, because the double negative on the constant gives a plus.
Questions students ask
How do I know which method to use to solve a quadratic equation?
Factorise if the quadratic factorises quickly and the answers are integers or simple fractions. Use the quadratic formula if the question asks for answers to 3 s.f., 2 d.p. or 3 d.p. Complete the square if it asks for the form (x + a)² + b or a minimum point, and use the graphical method only when told to use a graph.
Should I use substitution or elimination for simultaneous equations?
Use substitution when one variable has coefficient 1 and can easily be made the subject. Use elimination when neither variable has coefficient 1 or the coefficients already match: subtract if the matched coefficients have the same sign, and add if they have opposite signs. For one linear and one non-linear equation, always use substitution.
What does it mean when a quadratic equation has real roots?
“Real roots” includes equal roots, so the condition is b² − 4ac ≥ 0. Only the phrase “two distinct real roots” requires the strict inequality b² − 4ac > 0. Using the wrong sign wrongly includes or excludes the boundary value of k and loses the final answer mark.
How do you find when a quadratic is positive for all real x?
Two conditions must both hold: the coefficient of x² is positive (a > 0) and the discriminant is negative (b² − 4ac < 0). A negative discriminant alone is not enough, because a parabola opening downwards with b² − 4ac < 0 lies entirely below the x-axis. State both conditions, even when a > 0 is automatic.
In the full PDF
- •Further worked examples on the balance method, fractional coefficients and variables in the denominator
- •Worked substitution, elimination and line-meets-curve simultaneous equations
- •Full O-Level word problems on journey time, unit cost, area and consecutive numbers
- •Graphs of simultaneous equations, parabola roots and line–curve intersections, plus number-line diagrams
- •The remaining common mistakes and a summary at the end of every section
- •JC preview of fractional and modulus inequalities and graphical solving