How to Factorise a Quadratic Algebraic Equation
A quadratic equation
What is a scale factor?
When enlarging a shape or image, we use a scale factor to tell us how many times bigger we want each line/side to become. For example, if we enlarged a rectangle by scale factor 2, as in the image above, each side would become twice as long. If we enlarged by a scale factor of 10, each side would become 10 times as long.
The same idea works with fractional scale factors. A scale factor of 1/2 would make every side 1/2 as big (this is still called an enlargement, even though we have ended up with a smaller shape).
The same idea works with fractional scale factors. A scale factor of 1/2 would make every side 1/2 as big (this is still called an enlargement, even though we have ended up with a smaller shape).
A triangle being enlarged by scale factor 5
Enlarging with a scale factor of 5
In the diagram above, the left-hand triangle has been enlarged by a scale factor of 5 to produce the triangle on the right. As you can see, each of the three side lengths of the original triangle has been multiplied by 5 to produce the side lengths of the new triangle.
Scale factors with area
But what effect does enlarging by a scale factor have on the area of a shape? Is the area also multiplied by the scale factor?
Let's look at an example.
Let's look at an example.
Enlarging an area by scale factor 2
Enlarging an area by a scale factor
In the diagram above, we started with a rectangle of 3cm by 5cm and then enlarged this by a scale factor of 2 to get a new rectangle of 6cm by 10cm (each side has been multiplied by 2).
Look at what has happened to the areas:
Look at what has happened to the areas:
Original area = 3 x 5 = 15 cm^2
New area = 6 x 10 = 60 cm^2
The new area is 4 times the size of the old area. By looking at the numbers, we can see why this has happened.
The length and the height of the rectangle have both been multiplied by 2, hence when we find the area of the new rectangle, we now have two lots of x2 in there; hence the area has been multiplied by 2 twice, the equivalent of multiplying by 4.
More formally, we can think of it like this:
The length and the height of the rectangle have both been multiplied by 2, hence when we find the area of the new rectangle, we now have two lots of x2 in there; hence the area has been multiplied by 2 twice, the equivalent of multiplying by 4.
More formally, we can think of it like this:
After an enlargement of scale factor n:
New area = n x original length x n x original height
= n x n x original length x original height
= n^2 x original area.
So, to find the new area of an enlarged shape, you multiply the old area by the square of the scale factor.
This is true for all 2D shapes, not just rectangles. The reasoning is the same; area is always two dimensions multiplied together. These dimensions are both multiplied by the same scale factor; hence the area is multiplied by the scale factor squared.
This is true for all 2D shapes, not just rectangles. The reasoning is the same; area is always two dimensions multiplied together. These dimensions are both multiplied by the same scale factor; hence the area is multiplied by the scale factor squared.
Enlarging a volume by a scale factor
Enlarging a volume by a scale factor
What about if we enlarge a volume by a scale factor?
Look at the diagram above. We have enlarged the left-hand cuboid by a scale factor of 3 to produce the cuboid on the right. You can see that each side has been multiplied by 3.
The volume of a cuboid is height x width x length, so:
Look at the diagram above. We have enlarged the left-hand cuboid by a scale factor of 3 to produce the cuboid on the right. You can see that each side has been multiplied by 3.
The volume of a cuboid is height x width x length, so:
Original volume = 2 x 3 x 6 = 36 cm^3
New volume = 9 x 6 x 18 = 972 cm^3
By using division we can quickly see that the new volume is actually 27 times larger than the original volume. But why is this?
When enlarging the area we needed to take into account how two multiplied sides were both being multiplied by the scale factor, hence we ended up using the square of the scale factor to find the new area.
For volume it is a very similar idea, however this time we have three dimensions to take into consideration. Again, each of these is being multiplied by the scale factor, so we need to multiply our original volume by the scale factor cubed.
More formally, we can think of it like this:
When enlarging the area we needed to take into account how two multiplied sides were both being multiplied by the scale factor, hence we ended up using the square of the scale factor to find the new area.
For volume it is a very similar idea, however this time we have three dimensions to take into consideration. Again, each of these is being multiplied by the scale factor, so we need to multiply our original volume by the scale factor cubed.
More formally, we can think of it like this:
After an enlargement of scale factor n:
New volume = n x original length x n x original height x n x original width
= n x n x n x original length x original height x original width
= n^3 x original volume.
So to find the new volume of an enlarged 3d shape, you multiply the old volume by the cube of the scale factor.
Scale factor formula recap
If you are enlarging by a scale factor n:
Enlarged length = n x original length
Enlarged area = n^2 x original area
Enlarged volume = n^3 x original volume.
Comments
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