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Expanding something out like (x+2)(x+3) crops up everywhere in maths. Therefore it's a very important skill to develop.
There are many different ways of looking at it - smily faces, rainbows, claws - to name a few, and everyone has their favourite. However, all of them essential boil down to multipling everything in the first bracket with everything in the second bracket.
A good first way to look at it is to expand the first bracket to begin with. Taking our example at the top, we get (x+2)(x+3) = x(x+3) + 2(x+3).
We can then expand these as individual brackets: x2 + 3x + 2x + 6.
Finally, we can collect like terms to simplify: x2 + 5x + 6
And we're done. With practise, this becomes one of those things you do so often, you almost stop thinking about it. Until someone asks you what your favourite method is and then the arguments begin...see more
Easy! We first consider which number is in the square root. Then we look at it's factors:
- if any of factors are square we can immediately take their root out of the overall square root. Once all the square factors have been removed, we're as simple as we can get.
- if not, then there's not much to do as we're already simplified!
This is important as we can only add or subtract surds if they have the same number in the square root. Often we have to simplify the root before we can simplify larger, more complicated expressions.see more
Stars are born when gases at high pressure and temperatures begin to burn. They start life out as a big cloud of space junk and debris. Slowly, there's a bit of the cloud with more stuff in it than the other bits so it has a bit more gravitational pull and so slowly, the rest of the cloud gets pulled towards that general area.
But as more of the gas collects there, then there are even more gravitational forces acting so pressure increases and more elements continue to get pulled in. Pressure rises causes the temperature to rise and, once it's hot enough, hydrogen undergoes a nuclear fusion to form helium. This releases energy making it hotter and increases pressure. It also starts a chain reaction so more fusion occurs.
The star can then settle down to a long stable slog with gravity pulling things in balancing with the fusion pressure pushing things out. So there's not much growth and not much really happens. The star is known as a main sequence star. This is the stage our sun is at.
Eventually, however, all good things come to an end and there's no more hydrogen left. So fusion starts to happen with larger elements like helium. Stars like ours expand to become red dwarfs and become hotter. When our sun does this, it will expand to destroy the Earth.
When all the nuclear reactions are done, there's no outward pressure keeping the star big so gravity causes it to collapse into itself forming a white dwarf. Also, no reactions mean no energy so things begin to cool off a bit. And that’s that. Over billions of years, our sun has gone from a cloud to the sun we know and love and will eventually expand to destroy our planet before falling into itself and turning cold.see more