Quadratics

You will be able to

• Complete the square for ax² + bx + c and use a(x + p)² + q to state the vertex and the axis of symmetry.
• Use the discriminant b² − 4ac to decide between two distinct real roots, a repeated root and no real roots.
• Solve a quadratic equation exactly by the formula, factorising or completing the square, and solve a quadratic inequality from the sketch.

You will be able to

• Calculate the quadratic in x formed by substituting y = mx + k into y = ax² + bx + c, and its discriminant.
• Determine from the sign of b² − 4ac whether the line meets the curve at two points, touches it (a repeated root) or does not meet it.
• Deduce the set of values of k for each case, as an inequality, and the point of contact of a tangent.