Free · Year 2 of high school · 50 minutes · 15 questions · no penalty

Practice maths class test — Year 2: quadratic inequalities.

15 questions drawn at random from a bank of 25, on the block of the programme your teacher is finishing: every attempt is a different test. No penalty for wrong answers, so answer even if you are not sure. At the end you receive by email the correct answer to every question and your weak topics — with an account you see them straight away here on the page. The step-by-step solution unlocks with your first lesson.

A class test is the written test an Italian high-school class sits after each block of the programme; this practice version is multiple choice.

Test in ItalianThe questions of this simulator are in Italian, as in the real exam.

Test prepared by Gaetano Livornese · Aerospace engineer, Politecnico di Milano · maths and physics tutor · updated 13 October 2026

Programme covered by the test

Sign of the quadratic trinomial and the role of the discriminant
The cases Δ = 0 and Δ < 0, and a negative coefficient of x²
Inequalities with fractions and the sign table
Higher-degree and biquadratic inequalities
Systems of inequalities
Inequalities with absolute value
Inequalities with radicals and domain restrictions

Two worked examples from the test

Question 1 · Disequazioni di 2º grado, fratte, sistemi; modulo e irrazionali · Equazioni e disequazioni di 2° grado

Risolvi la disequazione $x^{2} - 7 x + 10 < 0$ studiando il segno del trinomio.

A $2 < x < 5$ correct answer
B $x < 2 \lor x > 5$
C $x \leq 2 \lor x \geq 5$
D $- 5 < x < - 2$

Solution

Passo 1 — Radici del trinomio. $Δ = 49 - 40 = 9 > 0 :$

$x = \frac{7 \pm 3}{2} \Rightarrow x_{1} = 2, x_{2} = 5$

Passo 2 — Disegno la parabola. $a = 1 > 0$ : concavità verso l'alto, taglia l'asse $x$ in $2$ e in $5$ .

Passo 3 — Leggo il segno.

$x < 2$ : parabola sopra l'asse $\to$ trinomio positivo
$2 < x < 5$ : parabola sotto l'asse $\to$ trinomio negativo
$x > 5$ : positivo

Passo 4 — Concludo. Serve $< 0$ , cioè la zona interna alle radici: $2 < x < 5$ .

Regola generale: il trinomio ha segno concorde con $a$ all'esterno delle radici e discorde all'interno. Chi risponde $x < 2 \lor x > 5$ ha usato la regola al contrario: è l'errore singolo più frequente di tutto il capitolo.

Question 2 · Disequazioni di 2º grado, fratte, sistemi; modulo e irrazionali · Equazioni e disequazioni di 2° grado

Risolvi $- x^{2} + 3 x + 10 > 0$ .

A $x < - 2 \lor x > 5$
B $- 5 < x < 2$
C $- 2 < x < 5$ correct answer
D $x < - 5 \lor x > 2$

Solution

Passo 1 — Rendo positivo il coefficiente di $x^{2}$ . Moltiplico per $- 1$ e cambio verso:

$x^{2} - 3 x - 10 < 0$

Passo 2 — Radici. $Δ = 9 + 40 = 49$ :

$x = \frac{3 \pm 7}{2} \Rightarrow x_{1} = - 2, x_{2} = 5$

Passo 3 — Segno. Ora $a > 0$ e il verso è $<$ : prendo l'intervallo interno.

Passo 4 — Concludo. $- 2 < x < 5$ .

Controllo veloce con $x = 0$ nella disequazione di partenza: $10 > 0$ , vero, e $0$ appartiene all'intervallo trovato. Chi risponde $x < - 2 \lor x > 5$ ha moltiplicato per $- 1$ senza invertire il verso.

In the online practice test you will find 13 more questions like these; the correct answer to every question arrives by email.

Prefer paper? Download the test as a PDF (15 questions, 4 pages · 79 KB, ready to print). You receive the solutions by email after taking it online.

Download the PDF

What a maths test on quadratic inequalities looks like

Quadratic inequalities in all three cases of the discriminant, at least one with fractions using a sign table, a system of inequalities, and generally one with absolute value or one with a radical.

The mistake that costs the grade

Choosing the intervals from memory. There is no “between the roots” rule that always works: it depends on the direction of the inequality and the sign of the coefficient of x². With Δ < 0 the trinomial has a constant sign and the inequality is either always true or always false, not “has no solutions” by definition.

How to use this practice test

Take it once cold, without revising and with the timer on: it is there to show you where you stand, not to look good. Then look at your weak topics, revise only those — usually two or three, not the whole chapter — and retake it after a few days. The 15 questions are drawn from a bank of 25, so the second attempt is not the same test and the scores are genuinely comparable.

Is the score not enough?

See which topics lost you the most marks, then read how to handle test anxiety — or we look at your results together: the diagnostic is free.

Book the free diagnostic call The other Year 2 tests