--- title: "Quadratic formula Practice Questions & Worked Answers" description: "Solve any quadratic equation, including the ones that will not factorise, and read the discriminant before you start. Free Years 10 to 11 practice with..." image: "https://nobleeducators.com/og-image.jpg" url: "https://nobleeducators.com/australia/practice/quadratic-formula" --- Your Australian school pathway · Noble Educators # Quadratic formula practice questions and worked solutions The quadratic formula solves every quadratic equation, including the many that will not factorise into neat brackets. Most students can recite it long before they can use it reliably, because the difficulty is not the formula itself but the bookkeeping around it: identifying a, b and c with their signs intact, and knowing what the discriminant is telling you before you commit to a method. Book a free demo WhatsApp now ![Student working through a lesson in a notebook beside a laptop](https://nobleeducators.com/sat-redesign/support-student.webp) Live, one-to-one learning Noble Educators 1 ## Diagnose before drilling The first session finds what is actually breaking down — missing content, a weak method, or lost confidence — because practising the wrong thing wastes a term. 2 ## Plan to the real calendar Lessons aim at the student's next school task — a SAC, assessment, trial, NAPLAN window or exam block — not a generic syllabus order. 3 ## Practise under matching conditions Timed where the task is timed, written where it is written, with the calculator and formula rules the real assessment uses. 4 ## Report so parents can see it After lessons, parents get plain-language notes: what was covered, what improved, and the next piece of work — no jargon, no guesswork. ## What the formula actually does Any quadratic can be written in the general form ax² + bx + c = 0, and the formula x = (−b ± √(b² − 4ac)) / 2a returns the values of x that satisfy it. Geometrically those values are where the parabola crosses the horizontal axis, which is why a quadratic can have two solutions, one, or none at all. The formula is derived by completing the square on the general form, and it is worth working through that derivation once even though you will never use it again directly. Students who have seen where the formula comes from make fewer sign errors with it, because the structure stops being arbitrary symbols to memorise. ## Reading the discriminant before you solve The expression under the square root, b² − 4ac, is called the discriminant, and it answers the question of how many real solutions exist before you do any of the arithmetic. A positive discriminant gives two distinct solutions, zero gives one repeated solution, and a negative value gives none in the real numbers. Checking it first is a genuine time saver in an exam. If the discriminant is negative you can state that there are no real solutions and move on rather than grinding through a formula that cannot produce an answer, and if it turns out to be a perfect square you know the quadratic factorises and there was a faster route available. The discriminant also appears in its own right in senior questions, particularly ones asking for the values of a parameter that make an equation have exactly one solution. Those questions are asking you to set the discriminant to zero and solve, which is only obvious once you think of it as a quantity rather than as part of a larger formula. ## Where marks are most often lost The dominant error is sign handling on c. In x² + 3x − 5 = 0 the value of c is negative five, so −4ac becomes positive twenty and the discriminant grows rather than shrinks. Writing a, b and c down explicitly as a labelled line before substituting costs five seconds and removes most of these mistakes. The second common loss is dividing only part of the numerator by 2a. The entire expression −b ± √(b² − 4ac) sits over the denominator, not just the square root term, and rushed working that drops the bracket produces an answer that looks plausible and is wrong. A third pattern appears when a is negative. Multiplying through by negative one first, so that a becomes positive, makes the arithmetic considerably safer and does not change the solutions. Examiners award method marks for a correct approach, so tidying the equation before you start is never wasted. ## When factorising is the better choice The formula always works, which is exactly why students reach for it when something quicker is available. If the discriminant is a perfect square the quadratic factorises over the integers, and factorising is usually faster and less error-prone than substituting three values into a formula with a square root in it. A quick habit that works well under time pressure is to look for two numbers that multiply to give ac and add to give b. If they are obvious within a few seconds, factorise; if they are not, go straight to the formula rather than continuing to hunt. Senior assessment rewards choosing efficiently, not just arriving at the answer. ## Practise it now 5 questions Choose an answer and the method appears straight away. Nothing is recorded and there is no time limit — work through them at whatever pace is useful, and read the working even on the ones you get right. 1. Question 1. Solve x² + 3x − 5 = 0 using the quadratic formula.x = (−3 ± √29) / 2x = (3 ± √29) / 2x = (−3 ± √11) / 2x = (−3 ± 29) / 2 2. Question 2. How many real solutions does 2x² − 4x + 5 = 0 have?NoneOneTwoInfinitely many 3. Question 3. The equation x² − 6x + 9 = 0 has a discriminant of zero. What does that tell you?There is one repeated solution, x = 3There are two solutions, x = 3 and x = −3There are no solutionsThe equation cannot be solved with the formula 4. Question 4. In the equation 5x² − 2x + 7 = 0, what is the value of c?7−25−7 5. Question 5. How many real solutions does 3x² + x − 2 = 0 have?TwoOneNoneIt depends on x Years 10 to 11 · Australian Curriculum v9.0 content descriptors AC9M10A01, AC9M10A02. Descriptor codes are published by ACARA and are quoted so you can check this practice against your school’s scope and sequence. Noble Educators is an independent tutoring provider and is not affiliated with ACARA. Common questions ## What families ask about this ### Do I have to memorise the quadratic formula? In most Australian senior courses, yes for the non-calculator components, though several states supply a formula sheet for external examinations. Check your own study design, because a formula you can look up still needs to be one you can apply quickly and without hesitation. ### What does it mean when the discriminant is negative? It means the equation has no real solutions and the parabola never crosses the horizontal axis. In courses that introduce complex numbers, usually specialist senior subjects, those solutions do exist but are written using the imaginary unit rather than as real values. ### Is the quadratic formula on the NAPLAN test? No. NAPLAN numeracy runs to Year 9 and covers linear relationships rather than solving quadratics with the formula, which is a Year 10 topic under the Australian Curriculum. Students who meet it earlier are usually in an extension or acceleration class. ### Why does my answer look different from the textbook's? Most often the two are equivalent and one has been simplified further, particularly where a surd can be reduced or a fraction cancelled. Substitute both back into the original equation and check whether each returns zero, which settles the question definitively. ## What sessions cost Every year level has a published per-session rate in AUD, with no lock-in contract. Year 1$12 AUD per session Year 7$18 AUD per session Year 12$26 AUD per session Booking a block reduces the per-session rate: 5% from 8 sessions, 8% from 12 sessions, 12% from 15 sessions, 15% from 20 sessions. At 12 sessions, Year 1 works out to $11.04 per session. The first lesson is free and there is no lock-in contract. [See the full rate card](https://nobleeducators.com/australia#pricing) ## How that compares Online 1-on-1, a private in-person tutor and a tutoring centre are genuinely different products, and the cheapest is not automatically the right one. These are the typical advertised rates for each in Australia. | Option | Typical cost per session | Format | | --- | --- | --- | | Noble Educators | $12–$34 | 1-on-1, live | | Private in-person tutor | $40–$100 | 1-on-1, in person | | Tutoring centre | $45–$90 | Small group, sometimes 1-on-1 | | Online marketplace tutor | $30–$70 | 1-on-1, live | Ranges describe typical advertised rates for each type of tutoring in Australia and are provided as general guidance only. They are not quotes from specific providers, and actual prices vary by tutor, year level, subject, location and session length. Compare current published pricing directly before deciding. [See the full cost comparison](https://nobleeducators.com/australia#compare) For parents ## Clarity before you book, visibility after every lesson. ## What to share before the first lesson The tutor match works from evidence, not a job title. Anything from the list below gives the first session a concrete starting point instead of a cold diagnostic. - The most recent school report or assessment result - The current task sheet, assessment notice or study design topic - Any teacher feedback, and the topic where confidence broke Send it through the enquiry form ## What you will see after every lesson Progress a parent cannot see is progress they cannot support. Reporting is part of the service, not an add-on you have to chase. - What the lesson covered and what changed, in plain language - The next piece of practice and why it comes next - A direct line to ask questions between lessons [See how the whole process works](https://nobleeducators.com/how-it-works) Your next step with Noble Educators ## Let’s find your way forward. Tell us your year level, school pathway and questions. Discuss a suitable tutor, lesson fees and a schedule that works for you. 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