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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 is always, sometimes or never true.

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.

# Log Lattice

##### Age 16 to 18 Challenge Level:

This resource is from Underground Mathematics.

Can you arrange some of these logarithms to complete the grid below? The values of the logarithms need to increase along the rows and down the columns. Try to do this without using a calculator.

$$\log_{3} 2 \quad \log_{4} 5 \quad \log_{2} 5 \quad \log_{3} 4 \quad \log_{3} 5 \quad \log_{5} 3$$
$$\log_{4} 2 \quad \log_{2} 4 \quad \log_{2} 3 \quad \log_{5} 2 \quad \log_{5} 4 \quad \log_{4} 3$$

• Which logarithms are bigger than $1$ and which ones are smaller?