Quadratics

• Completing the square: ax² + bx + c = a(x + p)² + q, with vertex (−p, q).
• The discriminant b² − 4ac decides the number of real roots: two, one repeated root, or none.
• Solve quadratic equations by factorising, completing the square or the formula.
• Solve quadratic inequalities by finding the roots, then sketching the parabola.
• Simultaneous equations (one linear, one quadratic): substitute the linear one into the quadratic.
• Equations that are quadratic in x², √x, tan x and so on: substitute u for that function.

a(x + p)² + q

Take out the coefficient of x² from the first two terms, halve the coefficient of x inside the bracket, and correct for the extra constant. Because (x + p)² ≥ 0, the completed square form shows the least (or greatest) value at once: the vertex of y = a(x + p)² + q is (−p, q), a minimum if a > 0 and a maximum if a < 0.

The discriminant of ax² + bx + c is b² − 4ac. It is the part of the quadratic formula under the square root, so its sign tells you how many real roots the equation ax² + bx + c = 0 has.

Equations by three methods, inequalities by a sketch

Factorise when you can see the factors quickly. Otherwise use the formula, or complete the square. For an inequality, solve the equation first to find the critical values, then sketch the parabola: for a positive x² coefficient the curve is below the axis between the roots and above it outside them.

Reduce everything to one quadratic

linear

An equation like x⁴ − 5x² + 4 = 0 is a quadratic in x². Let u = x² (or u = √x, u = tan x, …), solve for u, then go back to x and reject impossible values (for example √x cannot be negative).