This article is about quadratic equations and solutions. For more general information about quadratic functions, see Quadratic function. For more information about quadratic polynomials, see Quadratic polynomial. A quartic equation is a fourth-order polynomial equation of the form. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable. Monomial – is a polynomial with only one term.

Binomial – is polynomial with two terms.
Trinomial – is a polynomial with four or more terms.
Polynomial – is a polynomial with three terms.
Constant – a polynomial of degree zero.
Linear – a polynomial of degree one
Quadratic – a polynomial of degree two
Cubic – a polynomial of degree three
Quartic – a polynomial of degree four
Quintic – a polynomial of degree five
Degree – is the highest exponents or the highest sum of exponents of the variables in a term

A population is all the organisms that both belong to the same group or species and live in the samegeographical area. In ecology the population of a certain species in a certain area is estimated using the Lincoln Index. The area that is used to define a sexual population is such that inter-breeding is possible between any pair within the area and more probable than cross-breeding with individuals from other areas. Normally breeding is substantially more common within the area than across the border.[1] In sociology, population refers to a collection of human beings. Demography is a social science which entails the statistical study of human populations. This article refers mainly to human population.

The term community has two distinct commutive meanings: 1) Community usually refers to a social unit larger than a small village that shares common values. The term can also refer to the national community or international community, and, 2) in biology, a community is a group of interacting living organisms sharing...

...Jaquavia Jacques
Ms. Cordell
1st period
December 9, 2014
Quadratics is used to help to determine what is on a graph. There are many formulas that are used to put points on a graph to create parabolas. Parabolas are “U” shaped figures on a graph. Parabolas are examples of quadratics on a graph. Parabolas can be positioned up or down, which means if the arrows are going up it has a minimum point, and if the arrows are going down that means it has a maximum point. When graphing using the vertex formula: or the roots formula: whether the “a” is positive or negative helps identify if the graph has a maximum or minimum. Below are some examples and a visual representation of what a parabola looks like at its minimum and maximum points.
Example1:
Type Of Formula
Equation
“a” Positive or Negative
Max or Min
Vertex Formula
Negative
Minimum
Roots Formula
Positive
Maximum
There are three positions you may see a parabola when it is on a graph. Each position of the parabola determines the Nature of Roots. A parabola eithers has two real roots, one real root, or no real roots. The way you determine if a parabola has two real roots if the parabola “cuts” or cross” the x- axis in two places.
The roots formula also shows when a parabola has two real roots, which is the reason it is called the roots formula: because you can identify the two real roots by looking at the formula. To identify the roots you will set the. equal to...

...Mathematics With Equations
Jesse J. Oliver Jr.
Mathematics 126: Survey of Mathematical Methods
Professor Matthew Fife
Thursday, January 24, 2013
Ascertaining Mathematics With Equations
The abstract science of a number, quantity and space that can be studied in its very own right or as it may be applied to other disciplines and subject matters in several aspects, one considers to be that of mathematics. The problem of testing a given number for “primality” has been known to be proven by Euclid in ancient Greece that there are in fact infinitely many primes. In such relations, a mathematician will describe in example two projects that use prime numbers, composite numbers and the quadratic formula to solve equations.
From the projects section on page 397 of Mathematics in Our World, for Project One, the mathematician will work only equations ( a ) and ( c ), but complete each of the six steps (a-f) and for Project Two, the mathematician will select a minimal of five numbers including zero (0) as one number and the other four are to be two even and two odd numbers.
PROJECT ONE
Basically, the foundation of Project One originates from a thought provoking methodology for finding solutions to quadraticequations. These particular methodologies or rather the identifiable method became founded and created in the country of India. For project one, the mathematician...

...the nature of the roots of quadratics and cubic functions.
Part One
Case One
For Case One, the discriminant of the quadratic will always be equal to zero. This will result in the parabola cutting the axis once, or twice in the same place, creating a distinct root or two of the same root.
For PROOF 1, the equation y=a(x-b)2 is used.
PROOF 1
y = 3 (x – 2)2
= y = 3 (x2 – 4x + 4)
= y = 3x2 – 12x + 12
^ = b2 – 4ac
= (-12)2 – 4 x 3 x 12
= 144 – 144
= 0
The discriminant is equal to zero. The parabola touches the x-axis at (2, 0) and as a in the equation is a positive value, the parabola curves upwards.
For PROOF 2, the equation y=-a(x-b)2 is used.
PROOF 2
y = -3(x - 2)2
= y = -3 (x2- 4x + 4)
= y = -3x2 + 12x – 12
^ = b2 – 4ac
= 122 – 4 x (-3) x (-12)
= 144 – 144
= 0
The discriminant of the parabola is equal to zero. The parabola touches the x-axis at (2,0) and as a is a negative value, the parabola curves downwards.
For PROOF 3, the equation y=a(x-A)2 is used.
y = a (x – A)2
= y = a (x2 – 2Ax + A2)
= y = ax2 – 2aAx + aA2
^ = b2 – 4ac
= (- 2aA)2 – 4 x (a) x (aA2)
= 4a2A2 - 4a2A2
= 0
The discriminant of the parabola is equal to zero. The parabola touches the x- axis at (A, 0) and whether the parabola curves up or down is dependent on the whether A is a positive or negative value.
Case Two
In Case Two, the discriminant of the...

...Colleen Cooper
Solving QuadraticEquations
MAT 126 Survey of Mathematical Methods
Instructor: Kussiy Alyass
October 1,, 2012
Solving QuadraticEquations
Using correct methods to solve quadraticequations can make math an interesting task. In the paper below I will square the coefficient of the x term, yield composite numbers, move a constant term and see if prime numbers occur. I will use the text and the correct formulas to create the proper solutions of the two projects that are required and solve for the equations...
In project one, I will move the constant term to the right side of the equation and square the coefficient of the original x term and add it to both sides of the equation.
Solve for project one: Part A
X2 – 2x – 13 = 0.
4x*4 + 8 = 52
4x*4 + 8 +16 = 52 + 16
4x*4 + 8 + 16 = 68
2x + 4 = 12 2x + 12 + -12
2x = 4 2x = -6
X = 2 x = -3
Part C.
X2 + 12x – 64 + 0
4xsquared + 12x = 64 – 4 squared
16 + 48 = 64 – 16
64 = 48 + 4x squared
In project two, I will substitute numbers for x to see if prime numbers occur and then try to find a number for x when substituted in the formula, yields a composite number.
Project 2, Part A.
X2 – x + 41
8 x 8 + 41= 105 not a prime number
3 x 3 + 41 = 50 not a prime number
7 x 7 + 41 = 90 not a prime number
2 x 2 + 41 = 46 not a prime number
6 x 6 +...

...QuadraticEquationsEquationsQuadratic
MODULE - I
Algebra
2
Notes
QUADRATICEQUATIONS
Recall that an algebraic equation of the second degree is written in general form as
ax 2 + bx + c = 0, a ≠ 0
It is called a quadraticequation in x. The coefficient ‘a’ is the first or leading coefficient, ‘b’
is the second or middle coefficient and ‘c’ is the constant term (or third coefficient).
For example, 7x² + 2x + 5 = 0,
5
1
x² + x + 1 = 0,
2
2
1
= 0, 2 x² + 7x = 0, are all quadraticequations.
2
In this lesson we will discuss how to solve quadraticequations with real and complex
coefficients and establish relation between roots and coefficients. We will also find cube
roots of unity and use these in solving problems.
3x² − x = 0, x² +
OBJECTIVES
After studying this lesson, you will be able to:
• solve a quadraticequation with real coefficients by factorization and by using quadratic
formula;
• find relationship between roots and coefficients;
• form a quadraticequation when roots are given; and
• find cube roots of unity.
EXPECTED BACKGROUND KNOWLEDGE
• Real numbers
• QuadraticEquations with real coefficients.
MATHEMATICS
39...

...rectangle.
Find the values of a, b, c, d and e.
Length (m)
Width (m)
Area (m2)
1
11
11
a
10
b
3
c
27
4
d
e
(2)
3
(b)
If the length of the rectangle is x m, and the area is A m2, express A in terms of x only.
(1)
(c)
What are the length and width of the rectangle if the area is to be a maximum?
(3)
(Total 6 marks)
5.
(a)
Solve the equation x2 – 5x + 6 = 0.
(b)
Find the coordinates of the points where the graph of y = x2 – 5x + 6 intersects the x-axis.
Working:
Answers:
(a) …………………………………………..
(b) ..................................................................
(Total 4 marks)
4
6.
A picture is in the shape of a square of side 5 cm. It is surrounded by a wooden frame of width
x cm, as shown in the diagram below.
5 cm
x
l
The length of the wooden frame is l cm, and the area of the wooden frame is A cm2.
(a)
Write an expression for the length l in terms of x.
(1)
(b)
Write an expression for the area A in terms of x.
(2)
(c)
If the area of the frame is 24 cm2, find the value of x.
(4)
(Total 7 marks)
5
7.
(a)
Factorize the expression 2x2 – 3x – 5.
(b)
Hence, or otherwise, solve the equation 2x2 – 3x = 5.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
8.
A rectangle has dimensions (5 + 2x) metres and (7 – 2x) metres.
(a)
Show that the area, A, of the rectangle can be...

...
f(x)= 2x^2 + 8x + 5
Vertex: In order to find the vertex of the quadraticequation, begin by using the proper formula - b / 2(a) to find it. In the equation there is an (a) (b) (c) which is needed for the vertex equation. First look at the equation and determine what are the values of the three variable. In this case (a)= 2, (b)= 8, (c)= 5. Now plug them in properly into the vertex equation. * Notice in the vertex equation there is a negative sign!!! DO NOT FORGET! When plugged into the equation, it should state -8 / 2(2). Now use the proper steps and multiply the denominator by 2 as it implies resulting the denominator equalling 4. Now the vertex equation should be (-8 / 4) plus reducing would make it (-2). The (x) value of the vertex (-2, 0). To find the (y) value plug in the x-vertex back into the original equation f (x)= 2x^2+8x+5, resulting to be f(x)= 2(-2)^2+8(-2)+5. Use the order of operation (PEMDAS) to solve. * Do not forget that the exponent (^2) gets multiplied into (-2) not the product of 2(-2). After using those steps the f(x) or (y) value should equal -3, making the vertex (-2, -3).
x-intercept: The second step is finding the x-intercept (zeros, roots). Begin by using the quadratic formula to find the x-intercept. As stated in the process of finding the vertex, use the same variables and...