Sign of a quadratic: 'sign of a except between the roots'

The magic rule

Let $P(x) = ax^2 + bx + c$ with $a \neq 0$. Compute the discriminant $\Delta = b^2 - 4ac$.

If $\Delta > 0$: two roots $x_1 < x_2$

$P(x)$ has the sign of $a$ outside the roots, the opposite sign between the roots.

$x$	$-\infty$	$x_1$	$x_2$	$+\infty$
Sign of $P$	sign of $a$	$0$	sign of $-a$	$0$

If $\Delta = 0$: one double root $x_0$

$P(x)$ has the sign of $a$ everywhere, except at $x_0$ where $P = 0$.

If $\Delta < 0$: no roots

$P(x)$ has the sign of $a$ everywhere (never zero).

Typical example

$P(x) = -2x^2 + 4x + 6$.

$\Delta = 16 + 48 = 64 > 0$. Roots: $x_1 = -1$, $x_2 = 3$.

Here $a = -2 < 0$:

• $P > 0$ between $-1$ and $3$
• $P < 0$ elsewhere

Fast inequality

Solve $x^2 - 3x + 2 > 0$.

Roots: $1$ and $2$. Here $a = 1 > 0$.

⇒ $P > 0$ outside: $S = (-\infty, 1) \cup (2, +\infty)$.

Without computing the roots

If you spot a factoring quickly (e.g., $x^2 - 4 = (x-2)(x+2)$), use it directly.

If $\Delta$ isn’t a perfect square, leave $\sqrt{\Delta}$ in the roots.

Common mistakes

1. Forgetting the sign of $a$ ($-x^2 + 1$ is positive between $-1$ and $1$, not outside).
2. Confusing equation ($=0$) and inequality ($>0$).
3. Writing $\Delta = b^2 + 4ac$ (it’s $-4ac$).

?À toi de jouer

Sign of x² − 5x + 6 on ℝ?

A.Positive outside [2, 3], negative between 2 and 3B.Positive everywhereC.Negative between 5 and 6D.Negative everywhere

Valider