Generally, a quadratic equation has the following form: ax2+bx+c; with, c, b, and a being constant that are referred to as quadratic efficient, and x is a variable. In numerical stipulations, a quadratic equation is simply a polynomial-statement of a subsequent degree (Katz, & Barton, 2007). There are manifold methods of solving a quadratic equation, such as completing-the-square method; factoring; graphical method; employing quadratic formula; and Newton’s method. Superlatively, the rudimentary task of a prime numeral is generally set up by the arithmetic theorem.
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In mathematical terms, a prime-number is simply a natural numeral that has two distinct numeral divisors, i.e. the number itself and one (Stoppard, 1993). In this paper, someone will complete the weekly reading, i.e. up to Chapter 7 in the textbook entitled “Mathematics in Our World” that was written by Bluman G Allan; Thus, this paper aims at completing Project #1 and Project #2. In project 1, the paper will follow the example given by completing the six steps. In project 2, the paper will choose not less than five numbers, which can be odd, even numbers, or even zero, then substitute them in the formula to determine whether it yields a number that is composite or prime.
To answer this project, we ought to follow a fascinating method to solve quadratic equations. This method is believed to have originated from India, and it is referred to as the completing-the-square method (Bluman, 2005). Thus, this method will be employed in solving these equations:
- X2 – 2x-13=0;
- X2+12x-64=0; and
Project #1 Solution
We will follow the six steps as per the example given
- x2 – 2x = 13; Step 1, 4x2 – 8x = 52; then 4x2 – 8x + 4 = 56; from this,
We get (2x – 2)2 = 56; this implies that 2x-2 = √56 = 2√14, i.e. 2(x – 1) = 2√14
Simplifying we get x – 1 = √14; thus x = 1 + √14; Then 2x -2 = √56 = -2√14
From this 2(x – 1) = -2√14, which can be expressed as x – 1 = -√14;
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Therefore, x = 1 – √14
- X2– 4x = -3; Step 1, 4×2-16x =-12; then 4(x2– 4x) =-12; from this.
We get x2-4x =-3; this implies that x2-4x + 4 =1, i.e. (x – 2)2 = 1; simplifying we get
X-2 =1; thus x = 2 + √1 = 2 + 1 = 3; then x – 2 = -√1; from this x = 2 – √1, which can be expressed as x = 2 – 1 = 1
- x2 + 12x = 64; Step 1, 4x2+48x=256; then 4x2+48x+144 = 400; from this,
We get (2x + 12)2 = 400; this implies that 2x+12=√400; simplifying we get
2x + 12 = 20, x=4; or 2x + 12 = -20, and x= -16
- 2x2-3x=5; Step 1, x2-1.5x=2.5; then 4×2 – 6x = 10; step 3, 4x2 – 6x + 2.25 = 12.25;
Step 4, (2x – 1.5)2 = 12.25, step 5, 2x -1.5 = √12.25; and finally, step 6, 2x-1.5 = 3.5;
X= 2.5, or 2x – 1.5 = -3.5; x= -1
The searched formula that will be capable of yielding prime numbers is x2-x+41. This formula will be useful in this project, whereby at least 5 numerals for x will be selected and then substituted in this formula, so as to determine whether there is an occurrence of a prime number. Ideally, in this project, a numeral x that yields a composite numeral when substituted in the formula will also be found (Bluman, 2005). Thus;
Project #2 Solution
Let, P (x) =x2-x+ 41, lets plug-in the following values of x: x=1; x=2; x=5; x=10; x=12; x=20.
P (1) =12 -1 +41= 41, this is a prime number; P (2) = 22-2 + 41 = 43, this is a prime number;
P (5) =52-5+4 = 61, this is a prime number; P (7) = 72-7 41= 83, this is a prime number;
P (10) = 102-10+41=131, this is a prime number; P (12) =122-12+41=173, this is a prime number; and P (20) =202-20+41=421, this is also a prime number. In fact, this list will continue to yield prime numbers until the value of x equals to 40. For instance, if x=41 then
P (41) =412-41+41=1681, this is a composite numeral, but not a prime number.
The completing-the-square method can be employed in deriving the quadratic formula, to be capable of using the algebraic identity. Ideally, the majority of general notions that apply to the algebraic structure, in which multiplication, adding up and subtraction are defined, arise from prime numbers.
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Bluman, A. G. (2005). Mathematics In Our World. College of Allegheny: McGraw-Hill Publishers.
Katz, V. J., & Barton, B. (2007). History of Algebra. Educational and Mathematical Studies, 65 (2), 180-199.
Stoppard, T. (1993). Arcadia. London: Faber & Faber.