TMUA Quadratics (inc Hidden)

Quadratics and quadratic equations are a major feature in TMUA and an excellent knowledge of factorising quadratics, the quadratic formula (especially the discriminant b^2-4ac), drawing quadratics and such like is clearly essential. However most TMUA questions involve much more than this 'basic' knowledge - as such, questions involving quadratics can also be found under other topics such as 'Graphs and Graphical Solutions', 'Inequalities' and 'Roots of Equations'. But it is clear that TMUA loves quadratics (especially hidden quadratics where an equation  at first glance doesn't look anything like a quadratic but with a substitution becomes one). It is essential to be familiar with seeking them out and solving them. TMUA also loves 'unusual' quadratics (ie those not solved with traditional methods such as factorising or the quadratic formula) and sometimes difficult to see 'difference of two squares'.

For those so interested, we offer both one to one TMUA coaching on quadratics, as well as online group coaching via Zoom - 'TMUA Quadratics (inc Hidden)' is Session #8 of 20. Check our TMUA Group Coaching Schedule page for dates when these sessions are available - details of what we cover in each group session can be found on our TMUA Coaching Sessions Overview page. For more general information on our online sessions check out our  'Online Coaching FAQ'.

Feel free to contact us if you require more information regarding any of the above.

Free Revision Resources

The below list of questions links to short videos (teasers) on our YouTube channel which directly or indirectly test elements of what is required to succeed in answering TMUA questions on quadratics - they also provide a detailed solution to the question posed, together with tips and tricks and are arranged roughly in increasing order of difficulty. All the videos are taken from the Gresty Academy YouTube podcast 'A Crash Course in TMUA Must Know Maths' which contains hundreds of TMUA style questions and is great for revision.

• Teaser 413 - tests knowledge of finding undefined values for a quadratic fraction as denominator - Standard
• Teaser #261 - tests understanding of f(g(x)) and uses a neat trick to disentangle to a quadratic - Standard
• Teaser #473 - tests understanding of how to apply various transformations to a quadratic's vertex - Standard
• Teaser #22 - tests ability to find coordinates of intersection of a parabola and circle - Standard
• Teaser #20 - tests knowledge of vertex and focus of quadratic to find roots - Standard
• Teaser #77 - tests how to recalculate mean once a competitor drops out - Standard
• Teaser #136 - tests how to find the convergence value of a sequence by unearthing a hidden quadratic - Standard
• Teaser #435 - tests ability to use formula for tan(2x) to find tan(pi/12) - Medium
• Weekend Workout #16 - tests ability to see a hidden quadratic and then 'completing the square' to find a minimum  - Medium
• Teaser #497 - tests knowledge of when a cubic has exactly two distinct roots - Medium
• Teaser #478 - tests ability to find the maximum of an exponential expression involving a hidden quadratic - Medium
• Teaser #130 - tests finding maximum value of xy^2 given constraints - Medium
• Teaser #310 - tests knowledge of how to graph the modulus of a quadratic, and find the roots - Medium 
• Teaser #495 - tests how to rotate a quadratic about a non-origin point - Medium
• Weekend Workout #3 - tests ability to manipulate surds under square root by noticing a difference of squares - Medium
• Teaser #48 - tests 'I think of a number' word problem which becomes a quadratic - Medium
• Teaser #498 - tests how to establish when both axes are tangent to a circle - Medium
• Teaser #33 - tests when a cubic has only one real root - Medium
• Teaser #454 - tests how to solve a modulus equation involving a quadratic - Medium
• Teaser #28 - tests understanding of power rules to simplify a complex expression - Medium
• Teaser #494 - tests understanding of how to find positive solutions to a modulus equation - Medium
• Teaser #113 - tests ability to solve an equation involving sin and cos by converting to a quadratic - Medium
• Teaser #481 - tests ability to transform a parabola and calculate where the original and transformed curves cross - Medium
• Teaser #500 - tests understanding of when sqrt(x^2) = (sqrt(x))^2 - Medium
• Teaser #426 - tests understanding of how to solve an exponential equation with x being a power of a power of a power leading to a hidden quadratic - Medium
• Teaser #273 - tests ability to notice that a quadratic in x is actually a cos double angle formula - Medium
• Teaser #416 - tests knowledge of t-formulae to solve a trig equation and convert it to a quadratic in t- Medium
• Teaser #280 - tests knowledge of how to graph the modulus of a quadratic, and find the product of roots - Medium
• Teaser #37 - tests ability to sum a series of squares using difference of two squares trick - Medium
• Teaser #297 - tests ability to notice that a quadratic in x is actually a tan double angle formula - Medium
• Teaser #492 - tests ability to use graphical methods and a symmetry trick to solve a tough looking modulus equation - Quite Tough
• Teaser #489 - tests ability to integrate a product of two modulus - Quite Tough

TMUA Past Questions on Quadratics: SP p1 q2; SP p1 q6; SP p1 q9; SP p1 q11; SP p1 q17; SP p2 q3; SP p2 q7; 2016 p1 q8; 2016 p1 q11; 2016 p1 q13; 2016 p1 q16; 2016 p2 q3; 2016 p2 q15; 2017 p1 q5; 2017 p1 q14; 2017 p1 q19; 2017 p2 q14; 2018 p1 q4; 2018 p1 q15; 2018 p2 q9; 2018 p2 q20; 2019 p1 q1; 2019 p1 q2; 2019 p1 q7; 2019 p1 q11; 2019 p1 q12; 2019 p1 q13; 2019 p1 q15; 2019 p1 q20; 2019 p2 q19; 2020 p1 q5; 2020 p1 q6; 2020 p1 q8; 2020 p1 q9; 2020 p1 q10; 2020 p1 q11; 2020 p1 q15; 2020 p1 q17; 2020 p1 q19; 2020 p2 q1; 2021 p1 q4; 2021 p1 q5; 2021 p1 q6; 2021 p1 q8; 2021 p1 q12; 2021 p1 q14; 2021 p1 q18; 2021 p2 q15; 2022 p1 q1; 2022 p1 q2; 2022 p1 q9; 2022 p1 q12; 2022 p1 q13; 2022 p1 q15; 2022 p1 q16; 2022 p1 q17; 2022 p1 q20; 2022 p2 q1; 2023 p1 q2; 2023 p1 q7; 2023 p1 q11; 2023 p1 q20; 2023 p2 q10; 2023 p2 q18

Practice makes perfect - good luck in your studies!