TMUA Roots/Solutions of Equations

TMUA loves testing on roots of equations, everything from sums and products of roots (Vieta's formula for example) to finding equations with roots related to other roots, to seeing whether equations have roots, and if so how many etc (especially quadratics - see 'Quadratics (inc Hidden)'; cubics - see 'Cubics and Other Polynomials'; and solving using graphical methods - see 'Graphs and Graphical Solutions'). We also include in this topic solutions to various unusual equations which, if the question were rearranged, would be the roots (ie 'how many solutions are there to f(x)=g(x)?' is the same question as 'how many roots are there to f(x)-g(x)=0?') 

For those so interested, we offer both one to one TMUA coaching on roots/solutions of equations, as well as online group coaching via Zoom - 'TMUA Prerequisite Maths One' is Session #1 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 roots/solutions of equations - 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 #445 - tests understanding of when a cubic has exactly two distinct roots - Standard
• Teaser #392 - tests knowledge of the sum and product of the roots of a cubic equation - Standard
• Teaser #103 - tests understanding of symmetry of a quadratic and its roots - Standard
• Teaser #411 - tests transforming a cubic's roots from a, b, c to 2a-1, 2b-1 and 2c-1 - Standard
• Teaser #20 - tests knowledge of vertex and focus of quadratic to find roots - Standard
• Teaser #268 - tests how to find a quadratic whose roots are the square of another quadratic using a neat trick - Standard
• Teaser #129 - tests finding roots of a quartic given two roots - Standard
• Teaser #259 - tests finding a shared surd root of a line, quadratic and cubic - Standard
• Teaser #422 - tests ability to solve modulus equation by splitting into regions - Medium
• Teaser #310 - tests knowledge of how to graph the modulus of a quadratic, and find the roots - Medium 
• Teaser #280 - tests knowledge of how to graph the modulus of a quadratic, and find the product of roots using the fact the amended function is even - Medium
• Teaser #12 - tests ability to find domain of a log of a polynomial - Medium
• Teaser #33 - tests when a cubic has only one real root - Medium
• Teaser #476 - tests how to solve an inequality involving a polynomial over a polynomial - Medium
• Teaser #500 - tests understanding of when sqrt(x^2) = (sqrt(x))^2 - Medium
• Teaser #494 - tests understanding of how to find positive solutions to a modulus equation - Medium
• Teaser #497 - tests knowledge of when a cubic has exactly two distinct roots - Medium
• Teaser #252 - tests how to find a quadratic with transformed roots - Medium
• Teaser #197 - tests ability to find number of roots of y=e^(2x)+x-3  in an interval- Medium
• Teaser #502 - tests ability to split a function into its odd and even components - Medium
• Teaser #498 - tests how to establish when both axes are tangent to a circle - Medium
• Teaser #417 - tests ability to sketch the graph of the modulus of (ax+b)/(cx+d) and read off solutions to various equations - Quite Tough
• Teaser #482 - tests ability to find the second smallest positive solution to a trig equation - Quite Tough
• Teaser #477 - tests understanding of when a quartic has exactly two distinct roots - Quite Tough
• Teaser #322 - tests finding a pattern to sum a series of powers of a root of a cubic - Quite Tough
• Teaser #492 - tests ability to use graphical methods and a symmetry trick to solve a tough looking modulus equation - Quite Tough
• Teaser #298 - tests ability to find when ln(x) - ax^3 = 0 has no roots - Quite Tough
• Teaser #491 - tests understanding of all steps required  to correctly solve a log = log equation - Tough

TMUA Past Questions on Roots/Solutions to Equations: SP p1 q9; SP p1 q11; SP p1 q13; SP p1 q17; SP p2 q19; 2016 p1 q11; 2016 p1 q13; 2016 p2 q11; 2016 p2 q14; 2016 p2 q15; 2016 p2 q17; 2017 p2 q19; 2018 p1 q9; 2018 p2 q11; 2018 p2 q15; 2018 p2 q20; 2019 p1 q20; 2020 p1 q9; 2020 p1 q20; 2020 p2 q1; 2020 p2 q18; 2021 p1 q14;  2021 p2 q6; 2021 p2 q9; 2021 p2 q15; 2021 p2 q16; 2022 p1 q1; 2022 p1 q16; 2022 p1 q18; 2022 p2 q17; 2023 p1 q14; 2023 p2 q6; 2023 p2 q12; 2023 p2 q18; 2023 p2 q19

Practice makes perfect - good luck in your studies!