Study GuideYear 11Level 2 · StandardHSCVCE

Free Year 11 The Quadratic Formula and the Discriminant

An HSC/VCE-style study guide to solving any quadratic equation with the quadratic formula and using the discriminant to predict how many real solutions there are. Worked examples cover exact surd answers, equations with no real solutions, finding an unknown coefficient for equal roots, and proving a quadratic is always positive, with a six-question self-check and answers.

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Year: Year 11
Subject: Mathematics
Topic: Quadratics
Difficulty: Level 2 · Standard
Estimated time: 25 minutes
Curriculum: Australian Curriculum
Answers: Not applicable
Format: PDF (A4) + print

Students will practise

solving quadratic equations with the quadratic formula, including exact surd form
calculating the discriminant and interpreting its sign
finding unknown coefficients so that an equation has equal, distinct or no real roots
linking the discriminant to the number of x-intercepts of a parabola

Curriculum: Australian Curriculum. We show specific outcome codes only where they have been verified against the official curriculum document.

What's next?

Completed: The Quadratic Formula and the Discriminant

Study guide

How to use this study guide

Read it together first, pausing at each worked example to try the step before reading the answer.
Attempt the "Check yourself" questions at the end without looking back.
Then practise with a worksheet from the pathway above and finish with the topic test.

Common questions

Who is this study guide for?

Year 11 students (typically ages 16–17) working on quadratics. It is pitched at level 2 · standard.

Are the answers included?

This is a study guide, so there is no separate answer sheet; the 'Check yourself' questions include answers.

How long does it take?

About 25 minutes. Short, regular sessions work best: two or three a week beats one long one.

Do I need to sign up to download?

No. Click Download Free PDF and it opens immediately. It is free for personal, classroom and homeschool use.

What should we do next?

Try Year 11 Factorising and Solving Quadratics Worksheet — Level 3.

Year 11 · Mathematics · Quadratics

The Quadratic Formula and the Discriminant

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What you need to know

Any quadratic equation can be written as ax[sup 2] + bx + c = 0 with a ≠ 0. Factorising is fastest when it works, but the quadratic formula always works: x = [frac −b ± √(b[sup 2] − 4ac)/2a]. Identify a, b and c carefully (including their signs), substitute, and simplify. The whole of the top line is divided by 2a.

The part under the square root, Δ = b[sup 2] − 4ac, is called the discriminant. Its sign tells you how many real solutions the equation has before you solve it — and therefore how many times the parabola y = ax² + bx + c crosses the x-axis.

Discriminant	Number of real solutions	Graph of y = ax² + bx + c
Δ > 0	two distinct real solutions	crosses the x-axis twice
Δ = 0	one repeated (equal) solution, x =	touches the x-axis at the vertex
Δ < 0	no real solutions	does not meet the x-axis
Δ > 0 and a perfect square	two rational solutions (it would have factorised)	crosses the x-axis twice

Solve 2x² + 5x − 3 = 0

a = 2, b = 5, c = −3
Δ = 5² − 4(2)(−3) = 25 + 24 = 49 (a perfect square, so expect rational answers)
x = =
x = = [frac 1/2] or x = = −3
Check: 2()² + 5() − 3 = + − 3 = 0. Correct.

Solve x² − 4x + 1 = 0, leaving your answer in exact form

a = 1, b = −4, c = 1, so −b = 4
Δ = (−4)² − 4(1)(1) = 16 − 4 = 12
x =
Simplify the surd: √12 = √4 × √3 = 2√3
x = = 2 ± √3

Solve 3x² − 2x + 5 = 0

Δ = (−2)² − 4(3)(5) = 4 − 60 = −56
Δ < 0, so there is no real solution. (The parabola y = 3x² − 2x + 5 sits entirely above the x-axis.)

Find k so that x² + kx + 9 = 0 has equal roots

Equal roots means Δ = 0.
Δ = k² − 4(1)(9) = k² − 36
k² − 36 = 0, so k² = 36 and k = 6 or k = −6.
Check k = 6: x² + 6x + 9 = (x + 3)², a repeated root at x = −3. Correct.

For what values of m does 2x² − 3x + m = 0 have no real roots?

No real roots means Δ < 0.
Δ = (−3)² − 4(2)(m) = 9 − 8m
9 − 8m < 0, so 8m > 9 and m > [frac 9/8].

Show that x² + 3x + 5 is positive for all real x

Δ = 3² − 4(1)(5) = 9 − 20 = −11 < 0, so the parabola never touches the x-axis.
a = 1 > 0, so the parabola opens upwards and lies entirely above the x-axis.
Therefore x² + 3x + 5 > 0 for all real x. (Such a quadratic is called positive definite.)

Common mistake: sign errors with b and c. If the equation is x² − 4x + 1 = 0 then b = −4, so −b = +4 and b² = 16. If c is negative, −4ac becomes positive, so the discriminant gets bigger, not smaller. Another common mistake: dividing only part of the numerator by 2a. In , both the 4 and the 2√3 are divided by 2 to give 2 ± √3. Writing 4 ± √3 or 2 ± 2√3 is wrong.

Check yourself

Find the discriminant of x² + 6x + 9 = 0 and state the number of real solutions.
Solve x² + 3x − 10 = 0 using the quadratic formula.
Solve x² − 6x + 4 = 0, leaving your answer in exact form.
How many real solutions does 2x² + x + 1 = 0 have?
Find k so that x² − kx + 4 = 0 has equal roots.
For what values of m does x² + 4x + m = 0 have two distinct real roots?

Answers: 1. Δ = 36 − 36 = 0, one repeated solution (x = −3) 2. Δ = 49, x = , so x = 2 or x = −5 3. Δ = 20, x = = 3 ± √5 4. Δ = 1 − 8 = −7, so none 5. k² − 16 = 0, so k = ±4 6. 16 − 4m > 0, so m < 4

Preview — the PDF contains the full resource.

Worksheet
Year 11 Factorising and Solving Quadratics Worksheet — Level 3
Factorise monic quadratics and solve quadratic equations by factorising. A challenge set for students ready to stretch further. This Year…
Difficulty
Level 3 · Challenge
Time
15 min
Questions
10 questions
Answers
Answers included

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About this resource

Created by: Success Tutoring
Last reviewed: 1 October 2026
How it was made: Written by our team

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