Ch 1 · 1.1 Quadratics · A Level Mathematics 9709
Topic 1 of 8 in Pure Mathematics 1 — A Level Mathematics 9709 — 80 lessons.
Introduction
A ball thrown upwards, the profit from a price change, the area of a garden with a fixed fence: all follow a quadratic rule y = ax^2 + bx + c. Its graph is a parabola, and almost every Paper 1 question uses quadratic skills somewhere, often hidden inside a trigonometry or coordinate geometry question. (Syllabus section 1.1, Paper 1.)
Key ideas — Completing the square
Rewrite ax^2 + bx + c as a(x + p)^2 + q. Take out the factor a from the x terms, halve the coefficient of x, and correct for the extra constant:
🧑🏫 Teacher's explanation
Completing the square is like rearranging a messy room so that everything is in its perfect, symmetrical place. When we have a quadratic like ax^2 + bx + c, it is hard to see where the graph turns or where it hits the x-axis. By rewriting it into the form a(x+p)^2 + q, we create a 'perfect square' part that we can easily analyze.
📖 Definitions
🔍 Worked example — Standard
Find the coordinates of the vertex for the curve y = 2x^2 + 12x + 5.
🔍 Worked example — Exam-style
A quadratic function is given by f(x) = -x^2 + 4x + 7. State the coordinates of the maximum point and the equation of the line of symmetry.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — The discriminant
For ax^2 + bx + c = 0 the discriminant is b^2 - 4ac.
🧑🏫 Teacher's explanation
Imagine you are throwing a ball into the air. The height of the ball follows a quadratic path. The 'roots' of the equation are the moments when the ball is at ground level (y=0). The discriminant is like a diagnostic tool that tells you, without you having to actually solve the equation, whether the ball will ever hit the ground, whether it just barely grazes the ground, or whether it stays entirely above it.
📖 Definitions
🔍 Worked example — Easy
Determine the number of real roots for the equation x^2 - 5x + 6 = 0.
🔍 Worked example — Standard
Find the value of k for which the equation x^2 + kx + 9 = 0 has a repeated root.
🔍 Worked example — Exam-style
Show that the equation 2x^2 - 3x + 5 = 0 has no real roots.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Solving quadratic equations
Leave answers exact (surds) unless the question asks for decimals.
🧑🏫 Teacher's explanation
Solving a quadratic equation is essentially asking: "At what values of x does this parabola cross the x-axis?" While there are many ways to find these 'roots', the method you choose depends on how the equation looks. Factorising is like finding the puzzle pieces that multiply to make the expression, while completing the square is like rearranging the equation into a perfect square shape to isolate x.
📖 Definitions
🔍 Worked example — Easy
Solve x^2 - 5x + 6 = 0 by factorising.
🔍 Worked example — Standard
Solve 2x^2 + 8x + 3 = 0 by completing the square.
🔍 Worked example — Exam-style
Determine the number of real roots for 3x^2 - 4x + k = 0 if the equation has a repeated root. Find k.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
x + y = 5 x^2 - y^2 = 5 [4 marks]
Key ideas — Quadratic inequalities
Find the critical values (the roots), then sketch the parabola and read off where it is above or below the axis. For x^2 - x - 12 > 0: the roots are -3 and 4, the parabola opens upwards, so it is positive outside the roots:
🧑🏫 Teacher's explanation
Solving a quadratic inequality is like finding the 'safe zones' on a roller coaster track. A quadratic equation ax^2 + bx + c = 0 tells you exactly where the track hits the ground (the x-axis). A quadratic inequality, however, asks: 'In which regions is the track above the ground (>0) or below the ground (<0)?'.
📖 Definitions
🔍 Worked example — Easy
Solve the inequality x^2 - 5x + 6 < 0.
🔍 Worked example — Standard
Solve the inequality 2x^2 + 3x - 5 ≥ 0.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Simultaneous equations: one linear, one quadratic
Make one letter the subject of the linear equation and substitute into the quadratic. You get a quadratic in one letter; each root gives one point. Always find both coordinates of each solution.
🧑🏫 Teacher's explanation
Imagine you are looking at a map where a straight road (the linear equation) crosses a circular park (the quadratic equation). The points where the road enters and exits the park are the solutions to your simultaneous equations. To find these specific locations, you need to turn two equations into one.
📖 Definitions
🔍 Worked example — Easy
Solve the simultaneous equations: y = 2x + 1 and x^2 + y^2 = 10
🔍 Worked example — Standard
Solve x + y = 7 and x^2 - y = 5
🔍 Worked example — Exam-style
Find the coordinates of the points of intersection of the line y = x - 2 and the circle x^2 + y^2 = 4
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Equations that are quadratic in a function of x
Spot a quadratic in disguise and substitute:
🧑🏫 Teacher's explanation
Sometimes, a math problem looks much scarier than it actually is. Imagine you are looking at a complex machine with many gears, but if you look closely, you see that several gears are actually identical and just connected in a chain. If you could replace that whole cluster of identical gears with one single, simple gear, the machine becomes easy to understand.
📖 Definitions
🔍 Worked example — Easy
Solve x^4 - 13x^2 + 36 = 0
🔍 Worked example — Standard
Solve 2x - 5√(x) + 2 = 0
🔍 Worked example — Exam-style
Solve tan^2 θ - 3tan θ + 2 = 0 for 0 ≤ θ < π
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples — Example 1: vertex and least value
Express 2x^2 - 12x + 23 in the form a(x+p)^2 + q and state the coordinates of the vertex of y = 2x^2 - 12x + 23.
🔍 Worked example — Easy
Express x^2 - 6x + 11 in the form (x+p)^2 + q and state the coordinates of the vertex.
🔍 Worked example — Standard
Express 3x^2 + 12x + 7 in the form a(x+p)^2 + q and state the minimum value of the function y = 3x^2 + 12x + 7.
🔍 Worked example — Exam-style
A quadratic is given by y = -2x^2 + 8x + 5. Find the coordinates of the maximum point.
⚠️ Common errors
Worked examples — Example 2: equal roots
Find the values of k for which kx^2 + 4x + (k - 3) = 0 has equal roots.
🔍 Worked example — Easy
Find the values of c for which x^2 + 6x + c = 0 has equal roots.
🔍 Worked example — Standard
Find the values of k for which kx^2 + 4x + (k - 3) = 0 has equal roots.
🔍 Worked example — Exam-style
The equation 2x^2 + mx + 8 = 0 has equal roots. Find the possible values of m.
⚠️ Common errors
Worked examples — Example 3: line and circle
Solve x + y + 1 = 0 and x^2 + y^2 = 25.
🔍 Worked example — Standard
Solve x - y = 2 and x^2 + y^2 = 10.
🔍 Worked example — Exam-style
A line y = mx + c is a tangent to the circle x^2 + y^2 = 25 at the point (3, -4). Find the value of m.
⚠️ Common errors
Worked examples — Example 4: a disguised quadratic
Solve 2x - 7√(x) + 3 = 0.
🔍 Worked example — Easy
Solve x^4 - 5x^2 + 4 = 0
🔍 Worked example — Standard
Solve 2x - 7√(x) + 3 = 0
🔍 Worked example — Exam-style
Solve x^2 + 1/x^2 = 7
⚠️ Common errors
Common mistakes
Practice
Answers: (1) (x+4)^2 - 13; least value -13 (at x = -4). (2) k^2 - 36 > 0, so k < -6 or k > 6. (3) (2x+1)(x-3) < 0, so -tfrac12 < x < 3. (4) x^2 + x(2x-1) = 6 ⇒ 3x^2 - x - 6 = 0 ⇒ x = 1 ± √(73)/6, i.e. x ≈ 1.59,\ y ≈ 2.18 or x ≈ -1.26,\ y ≈ -3.51. (5) (x^2 - 1)(x^2 - 9) = 0, so x = ± 1 or x = ± 3. (6) -(x-3)^2 + 5; vertex (3, 5) is a maximum.