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Teaching & Learning Plans Quadratic Equations Junior Certificate Syll±us Maths.

Index. Section A: Section B: Section C: Section D: Section E: Section F: To solve quadratic equations using the formula (Higher Level only) Section G: To solve quadratic equations given in rational form.

Section A: Introduction. What is meant by finding the solution of the equation 4x + 6=14? Why is x =1 not a solution of 4x+6=14 How do we find the solution to 4x + 6 = 14? Remember to check your answer. What is 4 x 0 ? What is 5 x 0 ? What is 0 x 5 ? What is 0 x n ? What is 0 x 0? When something is multiplied by 0 what is the answer?.

Section A: Introduction. If xy = 0 what do we know about x and or y ? Compare (x - 3 )( x - 4) = 0, with xy = 0 and what information can we arrive at? If x - 3 = 0, what does this tell us about x ? If x - 4 = 0, what does this tell us about y? How do we check if x = 3 is a solution to (x - 3)(x - 4) = 0? How do we check if x = 4 is a solution to (x - 3)(x - 4) =0? Write in your copies in words what x = 3 or x= 4 means in the context of (x - 3)(x - 4) = 0. When an equation is written in the form (x - a)(x - b)=0, what are the solutions?.

Let x = 3 (3 - 1)(3 - 3) = 0 ( 2)(0) = 0 0 = 0 TRUE.

Section A: Student Activity 1. Note: It is always good practice to check solutions. The roots of a quadratic equation are the elements of its solution set. For example if x = 1,x = 2 are the root ⇒ = solution set. The roots of a quadratic equation are another name for its solution set . 1. If xy = 0, what value must either x or y or both have ? 2. Write in your own words what solving an equation means. 3. Solve the following equations: a. (x - 1) (x - 2) = 0 b. (x - 4) (x - 5) = 0 c. (x - 3) (x - 5) = 0 d. (x - 2) (x - 5) = 0 4. What values of x make the following statements true: a. (x - 2) (x - 5) = 0 b. (x - 4) (x + 5) = 0 c. (x - 2) (x + 4) = 0 5. Find the roots of (x - 4) (x + 5) = 0 ..

5. Find the roots of (x - 4) (x + 5) = 0. 6. Solve the equation (x - 3) (x + 2) = 0. Hence state what the roots of (x - 3) (x + 2) = 0 are. 7. Find a positive value for x that makes the statement (x - 4) (x + 2) = 0 true. 8. Solve the following equations: a. x (x - 1) = 0 b. x (x - 2) = 0 c. x (x + 4) = 0 9. a. These students each made at least one error, explain the error(s) in each case:.

Section A: Student Activity 1. How does the equation x (x - 5) = 0 differ from the equation ( x - 0 )( x - 5) = 0 ? Why are they the same? Hence what is the solution? What are the solutions of x (x - 6) = 0 ? Answer questions 8-11 on Section A : Student Activity 1 ..

8. Solve the following equations: a. x (x - 1) = 0 b. x (x - 2) = 0 c. x (x + 4) = 0 9. a. These students each made at least one error, explain the error(s) in each case:.

9. a. These students each made at least one error, explain the error(s) in each case: b . Solve each equation correctly showing all the steps clearly 10 . If x = 5 is a solution to the equation (x - 4) (x – b) =0what is the value of b? 11. Is x = 3 a solution to the equation (x - 3) (x - 2) Explain your reasoning. Solve this equation..

Section B: I ntroduction. . Lesson interaction.

-2 12 -1 6 0 2 1 0 2 0 3 2. abstract. abstract. Section B: I ntroduction.

-2 12 -1 6 0 2 1 0 2 0 3 2. Section B: I ntroduction.

Section B: Student Activity 2. a. Complete the following table: From the table above determine the values of x for which the equation is equal to 0. c. Solve the equation (x + 2) (x + 1) = 0 by algebra. What do you notice about the answer you got to parts b. and c. in this question? e. Draw a graph of the data represented in the above table. f. Where does the graph cut the x axis? What is the value of f (x) = (x + 2)(x + 1) at the points where the graph cuts the x axis? g. Can you describe three methods of finding the solution to (x + 2) (x + 1) = 0..

Section B: Student Activity 2. 2. Solve the equation (x - 1) (x - 4) = 0 a) by table, b) by graph and c) algebraically. 3. Write the equation represented in this table in the form (x−a)(x−b)=0..

4. The graph of a quadratic function f(x ) = ax 2 + bx +c is represented by the curve in the diagram. Find the roots of the equation f(x ) = 0 and so identify the function . 5. The graph of a quadratic function f(x) = ax 2 + bx +c is represented by the curve in the diagram. Find the roots of the equation f(x) = 0 and so identify the function..

6 . Where will the graphs of the following functions cut the x axis? a. f (x) = (x - 7) (x - 8) b. f (x) = (x + 7) (x + 8) c. f (x) = (x - 7) (x + 8) d. f (x) = (x + 7) (x - 8) 7 . For what values of x does (x - 7) (x - 8) = 0?.

. Section C: Introduction. Lesson interaction.

. Section C: Introduction. Lesson interaction.

Guide number 14 1x14 2x7. Solving q uadratic e quations x squared, x ’s, number equals zero.

Choose an appropriate method for factorising from the two below, you can hide the slide you do not want. Create more examples by duplicating the slide you wish and editing the numbers..

+6. . . 20:58.

+6. x 2. . 20:58. − 3 x. − 2 x. +6. x. x. − 3. − 2.

Section C: Student Activity 3. .

Section C: Student Activity 3. 5. A number is 3 greater than another number. The product of the numbers is 28. Write an equation to represent this and hence find two sets of numbers that satisfy this problem. 6. The area of a garden is 50cm 2 . The width of the garden is 5cm less than the breadth. Represent this as an equation. Solve the equation. Use this information to find the dimensions of the garden. 7. A garden with an area of 99m 2 has length x m. Its width is 2m longer than its length. Write its area in term of x. Solve the equation to find the length and width of the garden. 8. The product of two consecutive positive numbers is 110. Represent this as an algebraic equation and solve the equation to find the numbers..

. . . Section C : Introduction (II). The width of a rectangle is 5cm greater than its length. Could you write this in terms of x? If we know the area is equal to 36cm 2 , write the information we know about this rectangle as an equation. Solve this equation. What is the length and width of the rectangle? Is it sufficient to leave this question as x = 4 ?.

Section C: Student Activity 3. 9 . Use Pythagoras theorem to generate an equation to represent the information in the diagram. Solve this equation to find x. 10. One number is 2 greater than another number. When these two numbers are multiplied together the result is 99. Represent this problem as an equation and solve the equation. 11. Examine these students’ work and spot the error(s) in each case and solve the equation fully:.

Section C: Student Activity 3. . .

Section C: Student Activity 3. . .

Section D: Introduction. . Lesson interaction.

Choose an appropriate method for factorising from the two below, you can hide the slide you do not want. Create more examples by duplicating the slide you wish and editing the numbers..

. . . 20:58.

. 4x 2. . + 6 x. 20:58. − 10x. - 15. 2x. 2x. + 3.

. Section D: Introduction. Lesson interaction.

Section D: Student Activity 4. .

Section D: Student Activity 4. . +XE +trz.

Section D: Student Activity 4. . 0.5 -0.5.

. Section E: Introduction. Lesson interaction.

. Section E: Introduction. Lesson interaction.

Section E: Introduction. x x. x - y. . x 2 x x. x -y.

Section E: Student Activity 5. .

Section E: Student Activity 5. .

. . . Section F: Introduction. . . . . . Lesson interaction.

. Section F: Introduction. . Lesson interaction.

. . . Section F: Introduction. . . . . Complete the exercises in Section F:Student Activity 6..

Section F: Student Activity 6. . .

Section F: Student Activity 6. .

. Section G: Introduction. . . Lesson interaction.

. Section G: Introduction. . . Lesson interaction.

Section G: Student Activity 7. Solve the following equations:.