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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.Objectives of this module: be able to place a quadratic equation into standard form, be familiar with the zero-factor property of real numbers.

Overview

  • Standard Form of A Quadratic Equation
  • Zero-Factor Property of Real Numbers

Standard form of a quadratic equation

In Chapter ( [link] ) we studied linear equations in one and two variables and methods for solving them. We observed that a linear equation in one variable was any equation that could be written in the form a x + b = 0 , a 0 , and a linear equation in two variables was any equation that could be written in the form a x + b y = c , where a and b are not both 0. We now wish to study quadratic equations in one variable.

Quadratic equation

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

The standard form of the quadratic equation is a x 2 + b x + c = 0 , a 0.

For a quadratic equation in standard form a x 2 + b x + c = 0 ,

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

Sample set a

The following are quadratic equations.

3 x 2 + 2 x 1 = 0. a = 3 , b = 2 , c = 1.

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5 x 2 + 8 x = 0. a = 5 , b = 8 , c = 0.
Notice that this equation could be written 5 x 2 + 8 x + 0 = 0. Now it is clear that c = 0.

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x 2 + 7 = 0. a = 1 , b = 0 , c = 7.
Notice that this equation could be written x 2 + 0 x + 7 = 0. Now it is clear that b = 0.

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The following are not quadratic equations.

3 x + 2 = 0. a = 0. This equation is linear .

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8 x 2 + 3 x 5 = 0.
The expression on the left side of the equal sign has a variable in the denominator and, therefore, is not a quadratic.

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Practice set a

Which of the following equations are quadratic equations? Answer “yes” or “no” to each equation.

6 x 2 4 x + 9 = 0

yes

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4 x 3 5 x 2 + x + 6 = 8

no

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4 x 2 2 x + 4 = 1

yes

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2 x 5 x 2 = 6 x + 4

no

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9 x 2 2 x + 6 = 4 x 2 + 8

yes

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Zero-factor property

Our goal is to solve quadratic equations. The method for solving quadratic equations is based on the zero-factor property of real numbers. We were introduced to the zero-factor property in Section [link] . We state it again.

Zero-factor property

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. Algebraically, if a · b = 0 , then a = 0 or b = 0 , or both a = 0 and b = 0.

Sample set b

Use the zero-factor property to solve each equation.

If 9 x = 0 , then  x  must be  0.

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If 2 x 2 = 0 , then x 2 = 0 , x = 0.

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If 5 ( x 1 ) = 0 , then x 1 must be 0, since 5 is not zero.

x 1 = 0 x = 1

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If x ( x + 6 ) = 0 , then

x = 0 or x + 6 = 0 x = 6 x = 0 , 6.

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If ( x + 2 ) ( x + 3 ) = 0 , then

x + 2 = 0 or x + 3 = 0 x = 2 x = 3 x = 2 , 3.

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If ( x + 10 ) ( 4 x 5 ) = 0 , then

x + 10 = 0 or 4 x 5 = 0 x = 10 4 x = 5 x = 5 4 x = 10 , 5 4 .

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Practice set b

Use the zero-factor property to solve each equation.

( y + 6 ) ( y 7 ) = 0

y = 6 , 7

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( x + 5 ) ( 3 x 4 ) = 0

x = 5 , 4 3

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Exercises

For the following problems, write the values of a , b , and c in quadratic equations.

3 x 2 + 4 x 7 = 0

3 , 4 , 7

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2 y 2 5 y + 5 = 0

2 , 5 , 5

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3 a 2 + 4 a 1 = 0

3 , 4 , 1

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8 a 2 2 a = 0

8 , 2 , 0

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5 x 2 3 x + 9 = 4 x 2

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7 x 2 + 2 x + 1 = 6 x 2 + x 9

1, 1, 10

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3 x 2 + 4 x 1 = 4 x 2 4 x + 12

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5 x 7 = 3 x 2

3 , 5 , 7

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3 x 7 = 2 x 2 + 5 x

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0 = x 2 + 6 x 1

1 , 6 , 1

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For the following problems, use the zero-factor property to solve the equations.

3 ( k + 7 ) = 0

k = 7

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5 ( x + 4 ) = 0

x = 4

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y ( y 1 ) = 0

y = 0 , 1

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n ( n + 4 ) = 0

n = 0 , 4

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x ( x + 7 ) = 0

x = 7 or x = 0

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( y 4 ) ( y 8 ) = 0

y = 4 or y = 8

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( k 1 ) ( k 6 ) = 0

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( x + 5 ) ( x + 4 ) = 0

x = 4 or x = 5

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( y + 6 ) ( 2 y + 1 ) = 0

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( x 3 ) ( 5 x 6 ) = 0

x = 6 5 or x = 3

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( 5 a + 1 ) ( 2 a 3 ) = 0

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( 6 m + 5 ) ( 11 m 6 ) = 0

m = 5 6 or m = 6 11

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( 2 m 1 ) ( 3 m + 8 ) = 0

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( 4 x + 5 ) ( 2 x 7 ) = 0

x = 5 4 , 7 2

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( 3 y + 1 ) ( 2 y + 1 ) = 0

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( 7 a + 6 ) ( 7 a 6 ) = 0

a = 6 7 , 6 7

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( 8 x + 11 ) ( 2 x 7 ) = 0

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( 5 x 14 ) ( 3 x + 10 ) = 0

x = 14 5 , 10 3

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( 3 x 1 ) ( 3 x 1 ) = 0

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( 2 y + 5 ) ( 2 y + 5 ) = 0

y = 5 2

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( 5 m 6 ) 2 = 0

m = 6 5

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Exercises for review

( [link] ) Factor 12 a x 3 x + 8 a 2 by grouping.

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( [link] ) Construct the graph of 6 x + 10 y 60 = 0.
An xy coordinate plane with gridlines, labeled negative five and five with increments of one units on both axes.

A graph of a line passing through two points coordinates zero, six and five, three.

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( [link] ) Find the difference: 1 x 2 + 2 x + 1 1 x 2 1 .

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( [link] ) Simplify 7 ( 2 + 2 ) .

14 + 2 7

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( [link] ) Solve the radical equation 3 x + 10 = x + 4.

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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?
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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
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Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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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
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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.
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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
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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
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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
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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?
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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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