[i]This lesson will show you, how to use GeoGebra support to solve qudratic equalities, inequalities with parameter[/i]
[b]Age, educational level:[/b] 15-16 years, High School (upper secondary level, 10. schoolyear in Hungary)
[b]Educational goals:[/b] to understand solving qudratic equalities, inequalities with parameter, to strengthen IKT skills
[b]Equipments:[/b]
- minimal: computer with projector
- optimal: minimal + interactive board + one computer per student (for example netbooks)
[b]1. Introduction[/b]
Repetition: parametric equation, solving depend on parameter, number of solution of quadratic equation, relations between quadratic equation and quadratic function.
[i](This takes about 5'. Meanwhile you/sudents can turn on computer(s))[/i]
[b]2. Task [/b]
1. What kind of values of real parameter p have the two different real solutions of the following equation:
(p-2)x^2 - (p-5)x + 1 = 0
Solving with GeoGebra:
- make a slider (modify attributes; name: p, values: -10<=p<=10)
- type the equation at the command line
- modify p with the slider
Describe your detections!
Determine some p (for no solutions, 1 or 2 solutions)!
- define the left side of equations as function (f(x)=...)
Solve the task:
- D>0
- (-(p-5))^2 - 4(p-2)>0
- p^2 - 14p + 33 >0
- p<3 or p>11
Demonstrate this inequality with GeoGebra:
- type the inequality at command line (x^2 - 14 x + 33 > 0)
Describe your detections!
[i](This takes about 15'. The students can work alone - with netbooks -, or in pairs too.)[/i]
[b]3. Task [/b]
2. Describe the values of real parameter m, so that all real numbers must be solutions of the following inequality:
(m-1)x^2 + (2m+7)x + m-2 < 0
Solving with GeoGebra:
- make a slider (modify attributes; name: m, values: -10<=m<=10)
- type the equation at the command line
- modify m with the slider
Describe your detections!
Determine some m (where is no solutions, or all real numbers are solutions)!
- define the left side of equations as function (g(x)=...)
Formulate conditions according to all real numbers are solutions
Solve the task:
- D < 0 and
- (m-1) < 0 (must be together true)
- (2m+7)^2 - 4(m-1)(m-2) < 0
- 40m + 41 < 0
- m < - 41/40 and m < 1
- m < - 41/40
[i](This takes about 15'. The students can work alone - with netbooks -, or in pairs too.)[/i]
[b]4. Summary [/b]
The GeoGebra helps to
- estimate the solutions of equalities, inequalities
- understand the behavior of equalities, inequalities depends on parameter.
