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

Practice maths class test — Year 2: quadratic equations.

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

Pure (ax² + c = 0), no-constant-term (ax² + bx = 0) and complete equations
The discriminant and the number of solutions
Equations with fractions and domain restrictions
Relations between roots and coefficients
Equations with a parameter
The parabola as the graph of y = ax² + bx + c

Two worked examples from the test

Question 1 · Equazioni di 2º grado e parabola come grafico · Equazioni di secondo grado

Risolvi l'equazione pura: $4 x^{2} - 25 = 0$ .

A $x = \pm \frac{5}{4}$
B $x = \pm \frac{2}{5}$
C $x = \pm \frac{5}{2}$ correct answer
DNessuna soluzione reale

Solution

Passo 1 — Isolo il termine con $x^{2}$ .

$4 x^{2} = 25$

Passo 2 — Divido per il coefficiente di $x^{2}$ .

$x^{2} = \frac{25}{4}$

Passo 3 — Estraggo la radice con il doppio segno.

$x = \pm \frac{25}{4} = \pm \frac{5}{2}$

Un'equazione pura $a x^{2} + c = 0$ ha due soluzioni opposte quando $- \frac{c}{a} > 0$ . Chi risponde $\pm \frac{5}{4}$ ha diviso $5$ per $4$ invece di estrarre la radice anche del denominatore.

Question 2 · Equazioni di 2º grado e parabola come grafico · Equazioni di secondo grado

Risolvi usando la formula ridotta: $5 x^{2} - 14 x + 8 = 0$ .

A $x_{1} = - 2 \lor x_{2} = - \frac{4}{5}$
B $x_{1} = 10 \lor x_{2} = 4$
C $x_{1} = \frac{2}{5} \lor x_{2} = 4$
D $x_{1} = 2 \lor x_{2} = \frac{4}{5}$ correct answer

Solution

Passo 1 — Coefficienti. $a = 5$ , $b = - 14$ (pari, quindi si può usare la ridotta), $c = 8$ ; poniamo $\frac{b}{2} = - 7$ .

Passo 2 — Discriminante ridotto.

$\frac{Δ}{4} = (- 7)^{2} - 5 \cdot 8 = 49 - 40 = 9$

Passo 3 — Formula ridotta.

$x = \frac{7 \pm 9}{5} = \frac{7 \pm 3}{5}$

Passo 4 — I due valori.

$x_{1} = \frac{10}{5} = 2 x_{2} = \frac{4}{5}$

Nella formula ridotta si divide per $a$ , non per $2 a$ : chi lascia $10$ e $4$ ha dimenticato la divisione, chi trova valori negativi ha sbagliato il segno di $- \frac{b}{2} = + 7$ .

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 · 75 KB, ready to print). You receive the solutions by email after taking it online.

Download the PDF

What a maths test on quadratic equations looks like

Pure, no-constant-term and complete equations, one with fractions and domain restrictions, one with a parameter where you discuss the sign of the discriminant, and the relations between the sum and product of the roots. It often ends with the associated parabola to sketch.

The mistake that costs the grade

Applying the formula without first reducing the equation to standard form, and so with the wrong coefficients. In no-constant-term equations, taking out a factor and then dividing by x, losing the solution x = 0: you only divide by quantities you know are not zero.

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.

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