The algorithm was presented without proof to avoid others claiming the idea as their own. The method is clearly the same as the formula method of solution. He showed the solution of particular equations, for example: x2 - 10x = -9. He called the numbers by the names:
1 is "the square"
-10 is "the unknown"
-9 is "the absolute number"
Brahmagupta's work yields only one solution for each quadratic equation. However, by the 12th century, scholars were identifying both solutions. Problems were now being written to teach mental skills rather than to solve problems from astronomy.
x - 2.3) = x2- 3.8x + 3.45 the quadratic function can be form
This is a quadratic function.
When x is 3 this quadratic function is zero.
x2 - 4x + 3
x = 3
32 - 4 × 3 + 3
9 - 12 + 3
The Factor Theorem states that if a function of xis zero when x = 3, then (x - 3) is a factor of the function.
f(3) = 0 (x - 3) is a factor of f(x) = x2 - 4x + 3
The other factor can then be found to give the completely factorised form of the quadratic.
By multiplication it is found that the other factor is (x -1) and f(x) = (x - 3)(x -1)
The Factor Theorem states that if f(a) = 0 then (x - a) is a factor of the function f(x).
(x - a) is a factor of f(x) because f(a) = 0
The process can be used in reverse to find a quadratic function with particular properties. Find the function with factors (x - 3) and (x + 5)
The function with factors (x -3) and (x + 5) is
(x - 3)(x + 5)
x2 + 2x - 15
Sometimes functions with higher powers of x can be factorised by this method .
The process of obtaining the factors becomes longer and sometimes involves long-division.
f(x) = x3 - 12x2 + 47x - 60 f(5) = 0 (x - 5) is a factor.
Long division takes practise. Here is an example:
Divide: f(x) = x3 - 12x2 + 47x - 60 by the factor (x -5) to find the other factors
x cubed divided by x gives x squared multiply the factor (x -5) by x2 to give x3 - 5x2 subtract this from the first two terms and then divide the answer by x This gives -7x and after multiplication 7x2 + 47x Subtract to leave 12x which when divided by x gives 12
Long division is not essential. Other factors can be found using the Factor Theorem again. Factorising continues with the quadratic factor.
Therefore, the higher the R^2 (0?R^2?1), the closer the estimated regression equation fits the sample data. In this case, R^2 is 0.99624, which very close to unity. It indicates that the absolute equation fits the data very well according to R^2.
Some may use symmetry, some just trial and improvement, others may think about patterns of numbers. Extension work: Can the mean be 5? Can you get the mean to be between 6 and 7? * Obtain a mean of 5 and mode of 4 * Obtain a mean of 6 and mode of 3.
27=a5+b (2) 9=a (2)-(1) Substitute 'a' back into (1) 18=9x4+b 18=36+b b= -18 Substitute 'a' and 'b' back into second number equation S=9w-18 So we have the equations for the first and second numbers of the table. So now we put them together. The first number equation is solved by 4W-9 and then to that you had to times
Therefore a value added to x� translates the graph up or down on the y-axis. A different form of the quadratic function is to put it into a perfect square. A perfect square is an equation where the square root to the whole of one side is taken to get the answer on the other side.
(15X20-5X10)] We must then invert the corresponding signs of the integers using this shape: [ + - + ] [ - + - ] [ + - +] The final result is: [6 30 -120] [25 -40 -20 ] [-80 -40 250 ] We now know that A-1 is equal to the above result.
= x3-5x+0.1 f'(x) = 3x2-5 The iterative formula for the Newton-Raphson method is: xn+1 = xn- f(xn) f'(xn) My iterative formula is: xn+1 = xn- x3-5x+0.1 3x2-5 I will take my starting value of x (x1) to be 1.1. x1 = 1.1 x2 = -1.870073 x3 = -2.400053 x4 =
5 z4 = 52 + 1 = 26 and we see that this orbit tends to infinity, therefore 1 is not included in the Mandelbrot Set. If, however, the z0 valued 0 and the seed was 0, then the orbit would remain in a fixed position.
Over 160,000 pieces of student written work
Annotated by experienced teachers
Ideas and feedback to improve your own work
Want to read the rest?
Sign up to view the whole essay and download the PDF for anytime access on your computer, tablet or smartphone.