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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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
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John Reply
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Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
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David Reply
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David
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emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
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Adjanou
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Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
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Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
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Mohammed
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Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
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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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