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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

Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.
Widad Reply
can you send the book attached ?
Ariel
?
Ariel
What is economics
Widad Reply
the study of how humans make choices under conditions of scarcity
AI-Robot
U(x,y) = (x×y)1/2 find mu of x for y
Desalegn Reply
U(x,y) = (x×y)1/2 find mu of x for y
Desalegn
what is ecnomics
Jan Reply
this is the study of how the society manages it's scarce resources
Belonwu
what is macroeconomic
John Reply
macroeconomic is the branch of economics which studies actions, scale, activities and behaviour of the aggregate economy as a whole.
husaini
etc
husaini
difference between firm and industry
husaini Reply
what's the difference between a firm and an industry
Abdul
firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅
Abdulraufu
Suppose the demand function that a firm faces shifted from Qd  120 3P to Qd  90  3P and the supply function has shifted from QS  20  2P to QS 10  2P . a) Find the effect of this change on price and quantity. b) Which of the changes in demand and supply is higher?
Toofiq Reply
explain standard reason why economic is a science
innocent Reply
factors influencing supply
Petrus Reply
what is economic.
Milan Reply
scares means__________________ends resources. unlimited
Jan
economics is a science that studies human behaviour as a relationship b/w ends and scares means which have alternative uses
Jan
calculate the profit maximizing for demand and supply
Zarshad Reply
Why qualify 28 supplies
Milan
what are explicit costs
Nomsa Reply
out-of-pocket costs for a firm, for example, payments for wages and salaries, rent, or materials
AI-Robot
concepts of supply in microeconomics
David Reply
economic overview notes
Amahle Reply
identify a demand and a supply curve
Salome Reply
i don't know
Parul
there's a difference
Aryan
Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods
Israr
Hi Sir please how do u calculate Cross elastic demand and income elastic demand?
Abari
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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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