solving quadratic equations in one variable

Introduction

We studied before:

  • The equation of the   degree;       aX+b=0       ,  a≠0 
  • The equation of the  degree;       aX²+bX+c=0      ,    a≠0 
  • The degree of the equation → is the greatest power of the variabe.

Example: Determine the degree of the following equations;

  1. 2X+7=0.
  2. X²-5X+6=0.
  3. X³+1=0.

Solution;

  1.   degree.
  2. degree.
  3. degree.

 The quadratic equation has at most two solutions in R.

Task

Aids: Student book, board, colored marker, data show.

Aims: At the end of this lesson student would be able to;

  1. Solve the quadratic equation in one variable algebrically.
  2. Solve the quadratic equation in one variable graphically.
  3. Solve problems.

 You can refer to this video as a reference:

https://youtu.be/wM9FsXJslec

 

Process

Solving quadratic equation ( aX²+bX+c=0 ) in one variable algebrically: 

  1. By factorization:  we use the following property to find the two roots of equation;    if a, b ∈ R   ∃  ab=0   ,a=0   or   b=0.
  2. By the general formula:   .

Examples: Find in R the solution set of each of the following equations;

  1. X²-5X-6=0.
  2. X²+6X+9=0.
  3. X²-2X-6=0.

Solution;

  1. (X-6)(X+1)=0    [factorizing] ⇒∴ X-6=0  or  X+1=0  ⇒ ∴ X=6 ,X=-1 ⇒ ∴ The solution set is {6, -1}.
  2. (X+3)²=0    [factorizing perfect] ⇒∴ X+3=0  ⇒∴ X=-3  ⇒∴ The solution set is {-3} .
  3. [general formula]    a=1, b=-2, c=-6  ⇒   ⇒

  ⇒ ∴ The solution set is

Solving quadratic equation ( aX²+bX+c=0 ) in one variable graphically:

To solve the quadratic equation in one variable graphically, we follow the following;

  1. Put the equation on the form;  aX²+bX+c=0.
  2. Let F(X)=aX²+bX+c.
  3. Graph the function F.
  4. Determine the points of intersection of the curve with (X-axis), then the (X-coordinates) of these intersection points are the solution of the equation F(X)=0   i.e   aX²+bX+c=0.

According to that, we have three cases;

  • The curve intersects (X-axis) at two points:

 there are two solutions in R  ⇒ ∴ The solution set is {L, M}.

  • The curve touches  (X-axis) at one  point:

 there is a unique solution in R  ⇒ ∴ The solution set is {L}.

  • The curve  doesn’t intersect (X-axis):

 there is no solution in R   ⇒ ∴ The solution set is ∅.

 

 

Evaluation

Evaluation is done by noting the extent to which students participate in the class, cooperate with each other, and their ability to solve questions.

degree 1 3 5 final degree
display The display is unorganized, random, and does not serve the purpose. The display is unorganized, random, and does not serve the purpose. The offer is comprehensive, tidy and serves the purpose.  
teamwork and collaboration Work with individual effort, no cooperation between individuals, and tasks were not distributed among individuals. The work is organized, but not all people are involved, and tasks are not distributed effectively and collaboratively. The work is tidy and organized, and the tasks and tasks are distributed effectively and collaboratively.  
solve the educational problem The problem has not been properly resolved. One problem is solved, the other is irrational. Two problems have been solved correctly.  
         
         

Total degree=25 

Conclusion

We studied at this lesson the difference between the algebraically method and the graphically method.

The form of the quadratic equation is;    aX²+bX+c=0.

Teacher Page

quadratic equation page 9

algbriacal method page 10

graphical method page 11

exersices page 12