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10.8 Summary of key concepts

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This module is from Elementary Algebra</link>by Denny Burzynski and Wade Ellis, Jr. Methods of solving quadratic equations as well as the logic underlying each method are discussed. Factoring, extraction of roots, completing the square, and the quadratic formula are carefully developed. The zero-factor property of real numbers is reintroduced. The chapter also includes graphs of quadratic equations based on the standard parabola, y = x^2, and applied problems from the areas of manufacturing, population, physics, geometry, mathematics (numbers and volumes), and astronomy, which are solved using the five-step method.This module provides a summary of the key concepts from the chapter "Quadratic Equations".

Summary of key concepts

Quadratic equation ( [link] )

A quadratic equation is an equation of the form a x 2 + b x + c = 0 , where a ≠ 0. This form is the standard form of a quadratic equation.

a is the coefficient of x 2 .
b is the coefficient x .
c is the constant term.

Zero-factor property ( [link] )

If two numbers a and b are multiplied together and the resulting product is 0, then at least one of the numbers must be 0.

Solving quadratic equations by factoring ( [link] )

  1. Set the equation equal to 0.
  2. Factor the quadratic expression.
  3. By the zero-factor property, at least one of the factors must be zero, so, set each factor equal to zero and solve for the variable.

Extraction of roots ( [link] )

Quadratic equations of the form x 2 − K = 0 or x 2 = K can be solved by the method of extraction of roots. We do so by taking both the positive and negative square roots of each side. If K is a positive real number then x = K , − K . If K is a negative real number, no real number solution exists.

Completing the square ( [link] )

The quadratic equation a x 2 + b x + c = 0 can be solved by completing the square.
  1. Write the equation so that the constant term appears on the right side of the equal sign.
  2. If the leading coefficient is different from 1, divide each term of the equation by that coefficient.
  3. Find one half of the coefficient of the linear term, square it, then add it to both sides of the equation.
  4. The trinomial on the left side of the equation is now a perfect square trinomial and can be factored as ( ) 2 .
  5. Solve the equation by extraction of roots.

Quadratic formula ( [link] )

The quadratic equation a x 2 + b x + c = 0 can be solved using the quadratic formula.

a is the coefficient of x 2 .
b is the coefficient of x .
c is the constant term.

x = − b ± b 2 − 4 a c 2 a

Parabola ( [link] )

The graph of a quadratic equation of the form y = a x 2 + b x + c is a parabola.

Vertex of a parabola ( [link] )

The high point or low point of a parabola is the vertex of the parabola.

Questions & Answers

differentiate between demand and supply giving examples
Lambiv Reply
differentiated between demand and supply using examples
Lambiv
what is labour ?
Lambiv
how will I do?
Venny Reply
how is the graph works?I don't fully understand
Rezat Reply
information
Eliyee
devaluation
Eliyee
t
WARKISA
hi guys good evening to all
Lambiv
multiple choice question
Aster Reply
appreciation
Eliyee
explain perfect market
Lindiwe Reply
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
Ezea
What is ceteris paribus?
Shukri Reply
other things being equal
AI-Robot
When MP₁ becomes negative, TP start to decline. Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Kelo
yes,thank you
Shukri
Can I ask you other question?
Shukri
what is monopoly mean?
Habtamu Reply
What is different between quantity demand and demand?
Shukri Reply
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
ok
Shukri
how do you save a country economic situation when it's falling apart
Lilia Reply
what is the difference between economic growth and development
Fiker Reply
Economic growth as an increase in the production and consumption of goods and services within an economy.but Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
Abdisa Reply
any question about economics?
Awais Reply
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50. A,Calculate quantities of x and y which maximize utility. B,Calculate value of Lagrange multiplier. C,Calculate quantities of X and Y consumed with a given price. D,alculate optimum level of output .
Feyisa Reply
Answer
Feyisa
c
Jabir
the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema
Gsbwnw Reply
suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product
Abdureman
types of unemployment
Yomi Reply
What is the difference between perfect competition and monopolistic competition?
Mohammed
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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