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Powerful Quadratics
This comes in two parts, with the first being less fiendish than the second. It’s great for practising both quadratics and laws of indices, and you can get a lot from making sure that you find all the solutions. For a real challenge (requiring a bit more knowledge), you could consider finding the complex solutions.
Discriminating
You're invited to decide whether statements about the number of solutions of a quadratic equation are always, sometimes or never true.
Factorisable Quadratics
This will encourage you to think about whether all quadratics can be factorised and to develop a better understanding of the effect that changing the coefficients has on the factorised form.
Age 16 to 18
Challenge Level
Some equations involving powers or indices can be solved using logarithms... but not all.
Think about how you could go about solving the following equations. Sort them according to the tools or methods you would use.
| $3^x=81$ | $x^5=50$ | $3^x=43$ | $5^{2x}-5^x-6=0$ |
| $5^x+4^x=8$ | $5^x+2\times5^{1-x}=7$ | $3^{2x}-3=24$ | $2^{2x}-9\times2^x+8=0$ |
| $\sqrt{2x-3}=5$ | $5^x-x^5=3$ | $16^{\frac{3}{x}}=8$ | $\big(\frac{13}{16}\big)^{3x}=\frac{3}{4}$ |
You might find it helpful to have the equations printed on cards that you can rearrange as you sort them. They are available here: cards.pdf
Can you write some other equations to go in each of your sorting categories?
Underground Mathematics is hosted by Cambridge Mathematics. The project was originally funded by a grant from the UK Department for Education to provide free web-based resources that support the teaching and learning of post-16 mathematics.
Visit the site at undergroundmathematics.org to find more resources, which also offer suggestions, solutions and teacher notes to help with their use in the classroom.
NRICH is part of the family of activities in the Millennium Mathematics Project.