Topicm014582b47ca27a88_1528449000663_0Topic

The quadratic equations

Levelm014582b47ca27a88_1528449084556_0Level

Third

Core curriculumm014582b47ca27a88_1528449076687_0Core curriculum

III. Equations and inequations.

The student:

4) Solves quadratic equationsquadratic equationquadratic equations and inequations.

Timingm014582b47ca27a88_1528449068082_0Timing

45 minutes

General objectivem014582b47ca27a88_1528449523725_0General objective

Interpreting and handling information presented in the text, both mathematical and popular science, as well as in the form of graphs, diagrams, tables.

Specific objectivesm014582b47ca27a88_1528449552113_0Specific objectives

1. Communicating in English, developing mathematical and basic scientific‑technical and IT competence, forming of learning skills.

2. Solving quadratic equations.

Learning outcomesm014582b47ca27a88_1528450430307_0Learning outcomes

The student:

- solves quadratic equations,

- solves the word problems that require finding the roots of a quadratic equation.

Methodsm014582b47ca27a88_1528449534267_0Methods

1. Pyramid of priorities.

2. Situational analysis.

Forms of workm014582b47ca27a88_1528449514617_0Forms of work

1. Individual work.

2. Work in small groups.

Lesson stages

Introductionm014582b47ca27a88_1528450127855_0Introduction

The students work in groups. They remind the most important information about equations with one unknown using the pyramid of priorities method.

Procedurem014582b47ca27a88_1528446435040_0Procedure

The teacher informs the students that the goal of the lesson is solving quadratic equationsquadratic equationquadratic equations.

The students formulate the definition of the quadratic equation using the results of the group work.

Definition
The quadratic equation (with the unknown $x$) is the equation that can be transformed to the form $a{x}^2+bx+c=0$ where $a, b, c$ are known real numbers and $a\ne 0$.m014582b47ca27a88_1527752256679_0Definition
The quadratic equation (with the unknown $x$) is the equation that can be transformed to the form $a{x}^2+bx+c=0$ where $a, b, c$ are known real numbers and $a\ne 0$.

Discussion – how such equation can be solved. The students should note that solving quadratic equation is equivalent to determining x‑intercepts of the corresponding quadratic functionquadratic functionquadratic function.

The existence and the number of solutions of the quadratic equation $a{x}^2+bx+c=0$, where $a\ne 0$ depend on the sign of the discriminant $∆=b^2-4ac$.

The students determine together the content of the theorem on the number of the roots of the quadratic equationquadratic equationquadratic equation.

Theorem
The quadratic equation $a{x}^2+bx+c=0$, where $a\ne 0$:
- has no solution when ∆ < 0,
- has one solution $x_0=\frac{-b}{2a}$, when ∆ = 0,
- has two solutions $x_1=\frac{-b-\sqrt{∆}}{2a}$, $x_2=\frac{-b+\sqrt{∆}}{2a}$, when ∆ > 0.

The students determine together an algorithmalgorithmalgorithm of finding the roots of the quadratic equation.

[Illustration1]

The students solve the exercises together using the flowchart above

Task
Solve the equation:
a) 3x  - 2x - 7 = 0,
b) -36x  + 12x - 1 = 0,
c) 4x  + 3x + 77 = 0,

The students analyse the applet presenting the method of solving incomplete quadratic equationsquadratic equationquadratic equations without determining the discriminant and draw conclusions.

Task
[Geogebra applet]

Conclusions
- The equation $a{x}^2+bx+c=0$, when $a\ne 0$, $b\ne 0$ has two solutions: $x_1=0$, $x_2=-\frac{b}{a}$,
- The equation $a{x}^2+bx+c=0$ has no solution when $a·c>0$ , has two solutions:
$x_1=\sqrt{\frac{-c}{a}}$, $x_2=-\sqrt{\frac{-c}{a}}$ when  $a·c<0$.
- The equation $a{x}^2=0$ has one solution $x_0=0$

The students solve the exercises using the acquired knowledge.

Task
Solve the equation:
3xIndeks górny 2² - 2x - 7 = 0,
-36xIndeks górny 2² + 12x - 1 = 0,
4xIndeks górny 2² + 3x + 77 = 0,

Task
The sum of squares of three consecutive natural numbers is equal to 194. Find these numbers.m014582b47ca27a88_1527752263647_0The sum of squares of three consecutive natural numbers is equal to 194. Find these numbers.

Task
The perimeter of a rhombus is 116 cm. The lengths of the diagonals of this rhombus differ by 2 cm. Calculate the area of the rhombus.

An extra task
For what values of m the equation $x^2-2(m+2)x+4m+5=0$ has two roots fulfilling the condition $x_2-x_1=2$?

Lesson summarym014582b47ca27a88_1528450119332_0Lesson summary

Students perform consolidating exercises. Then they summarize together the lesson, formulating conclusions to be remembered:

The equation $a{x}^2+bx+c=0$, where $a\ne 0$,
- has no solution when ∆ < 0,
- has one solution $x_0=\frac{-b}{2a}$, when ∆ = 0,
- has two solutions $x_1=\frac{-b-\sqrt{∆}}{2a}$, $x_2=\frac{-b+\sqrt{∆}}{2a}$, when ∆ > 0.

Selected words and expressions used in the lesson plan

quadratic functionquadratic functionquadratic function

incomplete quadratic equationincomplete quadratic equationincomplete quadratic equation

quadratic equationquadratic equationquadratic equation

root of the quadratic equationroot of the quadratic equationroot of the quadratic equation

algorithmalgorithmalgorithm

number of solutions of an equationnumber of solutions of an equationnumber of solutions of an equation