Solving quadratics graphically

Description

The roots of an equation are the values of $x$ that make the equation equal to zero (i.e. where the graph crosses the $x$-axis).

Graphical method: plot $y = f(x)$ and identify where the graph crosses the $x$-axis. The $x$-values at these crossings are the roots.

$$f(x) = 0 \implies x = \text{roots}$$

For quadratics: find roots by factorising, using the quadratic formula, or completing the square.

Example: Find the roots of $y = x^2 - 5x + 6$. $$x^2 - 5x + 6 = 0 \implies (x-2)(x-3) = 0 \implies x = 2 \text{ or } x = 3$$

Trial and improvement (or iteration): for equations that cannot be solved exactly, use the graph to identify approximate roots, then narrow down using substitution or iteration.

The number of roots tells you about the nature of the equation — a quadratic can have 0, 1, or 2 real roots.

Common error: confusing roots (where $y = 0$) with $y$-intercept (where $x = 0$).

Links

Factorising quadratics Completing the square

Use the TABLE function to identify where $y$ changes sign — this brackets a root between two values.