📄 Cheat Sheets — Pure Mathematics 1

Key facts, formulas and common mistakes for 1 of 1 chapters

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8 key facts·2 formulas·6 common mistakes·12 definitions·6 exam tips

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Definitions

parabola: The shape of the graph of a quadratic function f(x) = ax^2 + bx + c.
vertex: The maximum or minimum point of a parabola.
stationary point: A point where the gradient of a curve is zero.
turning point: A point where the gradient of a curve is zero and the curve changes direction (from increasing to decreasing or vice versa).
roots: The solutions to the equation f(x) = 0 for a function f(x).
discriminant: The part of the quadratic formula underneath the square root sign, b^2 - 4ac.
two distinct real roots: A condition for a quadratic equation where the discriminant is positive (b^2 - 4ac > 0), resulting in two different real number solutions.
two equal real roots: A condition for a quadratic equation where the discriminant is zero (b^2 - 4ac = 0), resulting in exactly one real number solution (a repeated root).
no real roots: A condition for a quadratic equation where the discriminant is negative (b^2 - 4ac < 0), meaning there are no real number solutions.
points of intersection: The coordinates where two or more graphs meet.
line of symmetry: A vertical line that passes through the vertex of a parabola, dividing it into two mirror-image halves.
curve is positive: The range of x-values for which the y-values of a function are greater than zero (y > 0).

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Ch 01

Quadratic Formula

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Ch 01

Discriminant

Δ = b^{2} - 4 a c

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A quadratic equation is of the form ax^2 + bx + c = 0, where a ≠ 0.Ch 01
Completing the square transforms ax^2 + bx + c into a(x+p)^2 + q, revealing the vertex (-p, q).Ch 01
The quadratic formula x = (-b ± √(b^2 - 4ac)) / 2a solves any quadratic equation.Ch 01
The discriminant, Δ = b^2 - 4ac, determines the nature of the roots.Ch 01
Δ > 0 means two distinct real roots; Δ = 0 means two equal real roots; Δ < 0 means no real roots.Ch 01
The graph of a quadratic function is a parabola, with a vertex as its maximum or minimum point.Ch 01
To solve quadratic inequalities, find critical values and test regions or sketch the parabola.Ch 01
Points of intersection between a line and a quadratic curve are found by solving their equations simultaneously.Ch 01

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✗Don't divide by a variable (e.g., x) when solving equations, as this loses valid solutions.Ch 01
✗Remember to reverse the inequality sign when multiplying or dividing by a negative number.Ch 01
✗Don't forget to find the corresponding y-coordinates when solving simultaneous equations for points of intersection.Ch 01
✗Don't confuse 'two distinct real roots' (Δ > 0) with 'real roots' (Δ ≥ 0).Ch 01
✗Don't forget that for an equation to be quadratic, the coefficient of x^2 (a) cannot be zero.Ch 01
✗Don't confuse the vertex of a parabola with its x-intercepts (roots).Ch 01

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→Always check if factorisation is possible before resorting to the quadratic formula or completing the square.Ch 01
→When solving simultaneous equations, substitute the linear equation into the quadratic one to form a single quadratic equation.Ch 01
→For quadratic inequalities, sketching the parabola helps visualise the solution set and avoid errors.Ch 01
→Clearly state the nature of roots based on the discriminant value, using correct terminology (e.g., 'two distinct real roots').Ch 01
→When completing the square, ensure you correctly handle the coefficient 'a' if it's not 1.Ch 01
→Always verify your solutions by substituting them back into the original equations, especially for simultaneous equations.Ch 01