Quadratic functions, completing the square, the quadratic formula and the discriminant, simultaneous equations (linear and quadratic), and linear and quadratic inequalities.

What this dot point is asking

OCR wants you to work fluently with quadratic functions: solve them by factorising, by completing the square and by the formula; use the discriminant to determine the number and nature of roots; solve simultaneous equations where one is linear and one is quadratic; and solve linear and quadratic inequalities, presenting solutions in set notation where asked.

The answer

Solving quadratics

There are three standard methods. Factorising is fastest when it works. Completing the square writes $ax^2 + bx + c$ in the form $a(x + p)^2 + q$, which reveals the vertex and minimum or maximum value. The quadratic formula always works.

The discriminant

The discriminant is $\Delta = b^2 - 4ac$. It tells you the nature of the roots without solving:

• $\Delta > 0$: two distinct real roots.
• $\Delta = 0$: one repeated (equal) real root.
• $\Delta < 0$: no real roots.

This is a frequent source of "find the value of $k$" questions, where a condition on the roots becomes a condition on $\Delta$.

Simultaneous equations

When one equation is linear and one is quadratic, substitute the linear equation into the quadratic to get a single quadratic in one variable, solve it, then back-substitute.

Inequalities

For a linear inequality, solve as an equation, but reverse the sign when multiplying or dividing by a negative. For a quadratic inequality, find the roots, sketch the parabola, and read off the region. An upward parabola is positive outside the roots and negative between them.

Examples in context

Discriminant conditions

Disguised quadratics

Many equations are quadratics in disguise once you substitute. Equations in $x^4$ and $x^2$, in $\sqrt{x}$, or with a repeated bracket all reduce to a quadratic by a single substitution, which you then solve and reverse.

The vertex from completing the square

Completing the square does more than solve a quadratic: $a(x + p)^2 + q$ shows the vertex is at $(-p, q)$, the minimum value (for $a > 0$) is $q$, and the line of symmetry is $x = -p$. This is why completing the square, rather than the formula, is the right tool when a question asks for the minimum value of a quadratic or the range of a quadratic function.

Try this

Q1. Express $x^2 + 8x + 3$ in the form $(x + a)^2 + b$. [2 marks]

• Cue. $a = 4$, $b = 3 - 16 = -13$, so $(x + 4)^2 - 13$.

Q2. Solve $x^2 - 2x - 8 \le 0$. [3 marks]

• Cue. Roots $-2$ and $4$; between the roots, so $-2 \le x \le 4$.