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For your maths GCSE it is important that you understand the three main methods of solving quadtratics: factorisation, completing the square, and using the quadratic formula.
The first step for factorisation is to see if a common factor can be taken out, this is the easiest way of solving a quadratic.
2x2 + 4x = 0
In the case a factor of 2x can be taken out, making the equation look like this:
2x(x+2) = 0
This would then be solved by setting each part equal to zero,
2x = 0 and x+2=0.
Rearranging these equations gives us the final solutions of
x = 0 and x = -2.
If a common factor cannot be found, the next step is to try and put the equation into two brackets that are multiplied together.
Would be rewritten as:
^^^ when these brackets are multiplied out they give the original equation.
So in order to split the equation into two brackets we have to know which numbers are needed. The solution will be of the form
(ax+b)(cx+d) = 0, where a,b,c and d are integers.
So in our example,
a * c must equal 1 to give us the original 1x2.
a * d + b * c must be equal to 5 to give us 5x.
And b * d must be equal to 6 to give us our constant.
2. Completing the square:
To 'complete the square' of a quadratic, the initial equation is rewritten as a (x + constant) bracket squared minus another constant to give the same value as the starting equation. This is easier shown than explained with words.
x2+10x+20 = 0
First, the coefficient of x (the ten infront of the x) is halved, and this is the constant used in the bracket with x.
This gives us:
But we want (x+5)2 + a constant to be equal to x2+10x+30 = 0.
If we expand the squared bracket we get the x2 and the 10x that we need, but we get a +25, when we need +20.
To fix this we just take off another 5 after our squared bracket giving us a final equation of
(x+5)2 - 5 = 0
To solve for x we just add 5 to both sides and take the square root.
(x+5)2 = 5
x + 5 = +/- sqrt(5)
x = - 5 +/- sqrt(5)
3. Quadratic formula:
The last way of solving a quadratic is using the quadratic formula.
In the following example a, b, and c represent the integers in front of each part of the quadratic.
ax2 + bx + c = 0
To solve this using the quadratic formula, the integers just have to be subbed into the following equation:
x = [-b +/- sqrt (b2 - 4ac)] / 2a
This is quite difficult to type out but easy to actually use.
I hope this step by step guide of the methods of solving quadratic equations has been useful!
There are three methods of energy transfer that we need to learn: conduction, convection, and radiation.
Heat is thermal energy, and in solids it can be transferred by conduction. Heat is passed along from the hotter end of an object to the cold end by the particles in the solid vibrating. The hotter particles vibrate a lot and cause the particles next to them to vibrate as they gain heat energy too. Solids are heat conductors due to how tightly packed their particles are.
For example: When a saucepan is put on a hob, overtime the handle will get hot too. Due to conduction -> the heat from the bottom of the pan will cause the particles to vibrate and then cause all the surrounding particles to vibrate until the handle is hot too.
Fluids, that is both gases and liquids, can transfer heat energy by convection. It is easiest to explain this while thinking of an example:
Imagine a beaker of water being heated from the bottom. As the water particles at the bottom get hot, they expand and become less dense. This means they will rise to the top of the beaker, and other colder water particles will fall to replace them. After a while, the 'new' cold particles at the bottom will be heated and they will then rise to the top as they will be less dense. The water at the top which was first heated will have slightly cooled by then, so will sink down to the bottom, but then will be reheated and the same process will happen again.
This constant flow of the fluid due to the expansion / change in density of the particles is called a convection current. Over time all the fluid reaches a constant temperature.
Radiation is different to the other two processes as it doesn't require particles in its transfer of energy. Instead, infra-red radiation is a type of electromagnetic radiation. This means that the energy is transferred by waves rather than particles.
Radiation is how we feel the heat from the sun on Earth, as waves can pass through the vacuum of space where there are no particles.see more