Quadratics
Completing the square
- A quadratic is $ax^2 + bx + c$ (with $a \neq 0$).
- Completing the square rewrites it as $a(x+p)^2 + q$.
- This shows the vertex (turning point) at $(-p,\ q)$.
- Example: $x^2 - 6x + 5 = (x-3)^2 - 9 + 5 = (x-3)^2 - 4$ → vertex $(3, -4)$.
Practice
Completing the square, x² − 6x + 5 = (x − 3)² + q. What is the minimum value q?
(x − 3)² − 9 + 5 = (x − 3)² − 4, so q = −4 (the minimum value).
The discriminant
- The discriminant is $\Delta = b^2 - 4ac$. It counts the real roots:
- $b^2 - 4ac > 0$ → two distinct real roots,
- $b^2 - 4ac = 0$ → one repeated root,
- $b^2 - 4ac < 0$ → no real roots.
- The quadratic formula: $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
Practice
For x² + 4x + 1 = 0, what is the discriminant b² − 4ac?
b² − 4ac = 4² − 4(1)(1) = 16 − 4 = 12 (> 0, so two distinct real roots).
Practice
Solve x² − 5x + 6 = 0. What is the smaller root?
x² − 5x + 6 = (x − 2)(x − 3) = 0, so x = 2 or x = 3; the smaller is 2.
Practice
If b² − 4ac < 0, the quadratic equation has:
A negative discriminant means the parabola never crosses the x-axis — no real roots.
You've got it
Key idea
- complete the square: $a(x+p)^2 + q$, vertex at $(-p, q)$
- the discriminant $b^2 - 4ac$ counts real roots (+ two, 0 one, − none)
- solve with factorising, completing the square, or the quadratic formula