Pure Mathematics 1 · Quadratics
Carry out the process of completing the square for a quadratic polynomial and use a completed square form.
Find the discriminant of a quadratic polynomial and use the discriminant.
Solve quadratic equations, and quadratic inequalities, in one unknown.
Solve by substitution a pair of simultaneous equations of which one is linear and one is quadratic.
Recognise and solve equations in x that are quadratic in some function of x.
Understand the relationship between a graph of a quadratic function and its associated algebraic equation.
Use the relationship between points of intersection of graphs and solutions of equations.
Quadratic Formula
Used to solve quadratic equations of the form ax^2 + bx + c = 0.
Discriminant
Determines the nature of the roots of a quadratic equation ax^2 + bx + c = 0. If Δ > 0, two distinct real roots; if Δ = 0, two equal real roots; if Δ < 0, no real roots.
Students often incorrectly divide by a variable (e.g., x) when solving equations, losing valid solutions (e.g., in 3x^2 + 15x = 0, dividing by x loses x=0).
Students often forget to reverse the inequality sign when multiplying or dividing both sides of an inequality by a negative number.
Students often only find the x-coordinates when solving simultaneous equations for points of intersection, forgetting to find the corresponding y-coordinates.
Students often confuse 'two distinct real roots' (b^2 - 4ac > 0) with 'real roots' (b^2 - 4ac ≥ 0).
Students often forget that for an equation to be quadratic, the coefficient of x^2 (a) cannot be zero, especially when solving for conditions involving k in ax^2 + bx + c = 0.
Students often confuse the vertex of a parabola with its x-intercepts.