Factorising by Grouping Revision Quadratic Expressions Factorising Quadratic Equations Solving Solving Equations Revision Simultaneous Equations Taking out whats Common Difference of Two Squares ID: 477602
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Slide1
Topics in this Chapter
Factorising by Grouping (Revision)Quadratic Expressions (Factorising)Quadratic Equations (Solving)Solving Equations (Revision)Simultaneous EquationsTaking out what’s CommonDifference of Two SquaresChecklist
Mr. D. McCarthy
1Slide2
Factorise the following
x² + 6x + 2x + 12 (ii) x² + 3x + 4x + 12(iii) x² – 7x + 4x – 28 (iv) x² – 8x + 3x – 24
(x + 2)(x + 6)
Use this
(x + 4)(x + 3)
(x - 7)(x + 4)
(x - 8)(x + 3)Slide3
Factorise the following
x² - 8x - 2x + 16 (ii) x² - 12x - 4x + 48(iii) x² – 9x + 6x – 54 (iv) x² – 11x - 4x + 44
(x – 8)(x – 2)
(x - 12)(x - 4)
(x - 9)(x +
6
)
(x - 11)(x + 4)Slide4
Quadratic
Expression(Guide Number Method) x² + 14x + 48Guide Number = 48x² + 6x + 8x + 48(x + 6)(x + 8)Factors of 48:
1 x 482 x 24 3 x 16
4 x 12
6 x 8
4
Mr. D. McCarthySlide5
Quadratic
ExpressionFactorise the following1. x² + 10x + 21Guide Number = 21x² + 3x + 7x + 18(x + 3)(x + 7)
2. x² + 11x + 18Guide Number = 18
x² + 2x + 9x + 18
(x + 2)(x + 9)
Factors of 21:
1 x 21
3
x 7
Factors of 18:
1 x 18
2 x 93 x 6
5
Mr. D. McCarthySlide6
Quadratic
ExpressionFactorise the following:1. x² - 11x + 18Guide Number = 18x² - 2x - 9x + 18(x - 2)(x - 9) x² - 27x + 26 G.N. = 26
x² - x – 26x + 26(x – 1)(x – 26)
Factors of 18:
1 x 18
2 x 9
3 x 6
Factors of 26:
1 x 26
2 x 13
6
Mr. D. McCarthySlide7
Quadratic
ExpressionFactorise the following1. x² + 2x - 8Guide Number = -8x² + 4x – 2x – 8 (x + 4)(x – 2) x² - 6x – 16 G.N. =
-16 x² - 8x + 2x – 16
(x – 8)(x + 2)
Factors of -8:
1 x 8
2 x 4
Factors of –16:
1 x 16
2 x 8
4 x 4
7
Mr. D. McCarthySlide8
3 types of Quadratics
Factorise the following:x² + 11x + 28x² - 9x + 14
x² - 3x –
10
GN = 28
x² + 7x + 4x + 28
(x + 7)(x + 4)
GN = 14
x² - 7x -
2
x + 14
(x - 7)(x - 2)
GN = -10
x² - 5x
+
2
x - 10
(x - 5)(x + 2)
x² + 5x - 36
x² - x
-
12
x² + 3x –
28
GN = -36
x² + 9x - 4x –
36
(x + 9)(x - 4)
GN = -12
x² -
4
x
+
3
x –
12
(x – 4)(x + 3)
GN = -28
x² +
7
x
– 4
x –
28
(x
+
7)(x – 4)
8
Mr. D. McCarthySlide9
QUADRATICS
Algebra
Statistics
Trigonometry
Coordinate
Geometry
Functions
& Graphs
Complex
Numbers
Differentiation
9
Mr. D. McCarthySlide10
Spot the Difference
x² + 5x – 14 G.N = -14x² + 7x – 2x – 14(x + 7)(x – 2) Finished
G.N = -14
x² + 7x – 2x – 14 = 0
(x + 7)(x – 2) = 0
x + 7 = 0 or x – 2 = 0
x² + 5x – 14 = 0
Factors of -14
1 x 14
2 x 7
x = 2
x = - 7
G.N = 20
x² + 5x
+
4
x +
20
= 0
(x + 4)(x +
5
) = 0
x + 4 = 0 or x +
5
= 0
Solve:
x² + 9x + 20 = 0
G.N = 10
x² -
x – 10x +
10
= 0
(x –
1
)(x – 10) = 0
x – 1 = 0 or x – 10 = 0
Solve:
x² - 11x + 10 = 0
x = -5
x = - 4
x = 10
x = 1
Expression (factorise)
Equation (solve)
10
Mr. D. McCarthySlide11
Solve the following
x² + 8x + 15 = 0 G.N = 15x² + 5x + 3x + 15 = 0(x + 5)(x + 3) = 0 x + 5 = 0 or x + 3 = 0G.N = -22x² + 11x – 2x – 22 = 0
(x +11)(x – 2) = 0 x + 11 = 0 or x – 2 = 0
x² + 9x –
22
= 0
Factors of u ?
x = 2
x = - 11
G.N =
13
x² -
13
x
-
x +
13
= 0
(x
– 13
)(x
– 1
) = 0
x
– 13
= 0 or x
– 1
= 0
x² - 14x + 13 = 0
G.N =
-18
x² -
6
x +
3
x
– 18
= 0
(x –
6
)(x
+ 3
) = 0
x – 6 = 0 or x
= 3
= 0
x² - 3x – 18 = 0
x = 1
x = 13
x = -3
x = 6
x = - 5
x = - 3
11
Mr. D. McCarthySlide12
Solve the following equations
3x + 5 = 233x + 5 – 5 = 23 – 5 3x = 18³x/₃ = ¹⁸⁄₃x = 6
3x - 2y = 16
3x + 2y = 16
Cannot solve!
You cannot solve an equation with an x and a y in it.
What if you were told y = 4?
3x - 2(4) = 16
3x – 8 = 16
3x – 8 + 8 = 16
+ 83x = 24³x/
₃ = ²⁴⁄₃
x= 8
12
Mr. D. McCarthySlide13
Solve the following
4x + 6y = 22 when y = 34x + 6(3) = 224x + 18 = 224x + 18 – 18 = 22 – 18 4x = 4x = 1 5x – 2y = 20when y = 5
5x – 2(5) = 205x – 10 = 205x – 10
+ 10
= 20
+ 10
5x = 30
x = 6
3
x + 7y = 40
when x = 4
3(4) + 7y = 40
12
+ 7y = 40
12
– 12
+ 7y = 40
– 12
7y = 28
y = 4
5x
+
2
y = 9
when y = -3
3(4) + 7y = 40
12
+ 7y = 40
12
– 12
+ 7y = 40
– 12
7y = 28
y = 4
13
Mr. D. McCarthySlide14
Solve the following
(i) 4x = 20(ii) 3x = 12(iii) 5y = 30(iv) 8x = 12(v) 6x = 24(vi) 10x = 50
x = ²⁰⁄₅
x = 4
x = ¹²/₃
x = 3
x = ³⁰/₅
x = 6
x = ¹²/₈
x = ³/₂
x = ²⁴/₆
x = 4
x = ⁵⁰/₁₀
x = 5
14
Mr. D. McCarthySlide15
Simultaneous Equations
Solve for x and y3x + y = 142x – y = 65x + 0 = 205x = 20x = ²⁰⁄₅x = 4What do you add to the following to make them 0?(i) 3x
(ii) 6y
(iii) -3x
(iv) -y
- 3x = 0
- 6y = 0
+ 3x = 0
+ y = 0
x = 4
3x + y = 14
3(4) + y = 14
12 + y = 14
12 – 12 + y = 14 – 12
y = 2
15
Mr. D. McCarthySlide16
Solve for x and y
3x – 2y = 104x + 0 = 164x = 16x = ¹⁶⁄₄x = 4 -2x – y = -120 + y = 2y = 2
2x + 2(2) = 142x + 4 = 14
x + 2y = 6
4 + 2y = 6
4 – 4 + 2y = 6 – 4
2y = 2
y = 1
2x + 2y = 14
16
Mr. D. McCarthySlide17
Solve for x and y
x + 2y = 104x – 2y = 105x + 0 = 205x = 20x = ²⁰⁄₅x = 4 3x + y = 13-2x + y = - 2
5x + 0 = 155x = 15x = ¹⁵⁄₅
x = 3
3(3) + y = 13
9 + y = 13
9 – 9 + y = 13 – 9
y =4
x + 2y = 10
4 + 2y = 10
4
– 4 + 2y = 10 – 4
2y = 6
y = 3
3x + y = 13
(+) (-) (+)
17
Mr. D. McCarthySlide18
Exericse
2x + y = -2 -2x - 6y = -180 + 5y = -205y = - 20 5y/5 = -20/5y = -4
2x + y = -2
2x + (-4) = -2
2x – 4 = -2
2x – 4
+ 4
= -2
+ 4
2x = 2x = 1
12x - 3y = -27 2x - 3y = -710x + 0 = -2010x = - 20
10x/10 = -20/
10
x = -2
12x - 3y = -27
12(-2) - 3y = -27
-24 - 3y = -27
-24
+ 24
- 3y = -27
+ 24
-3y = -3
x = -1
(-) (+) (+)
7x - 2y = -2
3x + y = 14
7x – 2y = -2
6x + 2y = 28
13x + 0 = 26
13x = 26
13x
/
13
=
26
/
13
x = 2
7(2) - 2y = -2
14 - 2y = -2
14
– 14
- 2y = -2
– 14
-2y = -16
y = 8
7x - 2y = -2
( x 2)
18
Mr. D. McCarthySlide19
Solve the following
2x + 3y = 11 when y = 32x + 3(3) = 112x + 9 = 112x + 9 – 9 = 11 – 9 2x = 11 – 92x/
2 = 2/
2
x = 1
2x – 6y = 20
when y = -4
2x – 6(-4) = 20
2x + 24 = 20
2x + 24 – 24 = 20 – 24
2x = -42x/2 = -4
/2x = -2
19
Mr. D. McCarthySlide20
Exercise
2x + y = 19 2x + 6y = 340 - 5y = -15-5y = - 15 -5y/-5 = -15/-5y = 3 2x + y = 19
2x + 3 = 19
2x + 3
– 3
= 19
– 3
2x = 16
x = 8
(-) (-) (-)
3
x +
y = 5
5
x
–
4y = -3
12
x +
4
y = 20
5
x –
4
y = -3
17
x + 0 = 17
17x
/
17
=
17
/
17
x = 1
3(1) + y = 5
3 + y = 5
3
– 3
+ y = 5
– 3
y = 2
3x + y = 5
( x 4)
20
Mr. D. McCarthySlide21
Solve the following
2x + 3y = 11 when y = 32x + 3(3) = 112x + 9 = 112x + 9 – 9 = 11 – 9 2x = 22x/2
= 2/2
x = 1
2x – 6y = 20
when y = -4
2x – 6(-4) = 20
2x + 24 = 20
2x + 24
– 24 = 20 – 24 2x = -4
2x/2 = -4/
2x = -2
21
Mr. D. McCarthySlide22
Solve the following
2x + 3y = 11 when y = 32x + 3(3) = 112x + 9 = 112x + 9 – 9 = 11 – 9 2x = 11 – 92x/
2 = 2/
2
x = 1
2x – 6y = 20
when y = -4
2x – 6(-4) = 20
2x + 24 = 20
2x + 24 – 24 = 20 – 24
2x = -42x/2 = -4
/2x = -2
22
Mr. D. McCarthySlide23
Exercise
x + 2y = 133x – 5y = 63x + 6y = 393x – 5y = 60 + 11y = 3311y/11 = 33/11y = 3
4x – 5y = 172x – y = 7
4x – 5y = 17
4x – 2y = 14
0 – 3y = 3
–3y
/
-3 = 3/-3
y = -1
(-) (+) (-)(-) (+) (-)
x + 2y = 13
x + 2(3) = 13
x + 6 = 13
x + 6
– 6
= 13
– 6
x = 7
(x 3)
4x – 5y = 17
4
x
-
5
(-1) = 17
4x + 5 = 17
4x + 6
– 5
= 17
– 5
4x = 12
x = 3
(x 2)
23
Mr. D. McCarthySlide24
Exercise
Solve the followingx² - 4x – 21 = 01. GN = -212. Factors1 x 213 x 7x² - 7x + 3x – 21 = 0 (x – 7)(x + 3) = 0 x – 7 = 0 x + 3 = 0
7. x = 7 x = -3
x² - 11x
+
18
= 0
1. GN = -21
2.
Factors
1 x 21
3 x 7
x² - 7x + 3x – 21 = 0
(x – 7)(x + 3) = 0
x – 7 = 0 x + 3 = 0
7. x = 7 x = -3
24
Mr. D. McCarthySlide25
What do we do here
?x² - 8x = 0Take out what’s common!x(x – 8) = 0 x = 0 x – 8 = 0 x = 83x² + 15x = 03x(x + 5) = 0 3x = 0 x + 5 = 0
x = 0
x + 5 = 0
x = -5
x² +
7
x = 0
x(x – 8) = 0
x = 0
x – 8 = 0
x = 8
4x²
- 12x = 0
4
x(x – 3) = 0
4x = 0
x – 3 = 0
x = 0
x = 3
25
Mr. D. McCarthySlide26
Difference of two squares
Describe these equations:x² - 36 = 0( x + )( x - ) = 0x + 6 = 0 x – 6 = 0 66x = -6
x = 6
x² - 16 = 0
( x + )( x - ) = 0
x + 4 = 0 x – 4 = 0
4
4
x = -4
x = 4
x² - 25 = 0
( x + )( x - ) = 0
x + 5 = 0 x – 5 = 0
5
5
x = -5
x = 5
x² - 9 = 0
( x + )( x - ) = 0
x + 3 = 0 x – 3 = 0
3
3
x = -3
x = 3
26
Mr. D. McCarthySlide27
For Test
Page 269Q3(a)(b)Q4(a)(b)Q5 (a)(b)Also 28 – 39 page 261IF YOU CAN DO MOST OF THESE YOU WILL HAVE NO PROBLEM WITH THE TEST! 27Mr. D. McCarthySlide28
Checklist fo
r this chapterQuadratics(i) x² + 9x + 14 = 0(ii) x² - 12x + 20 = 0(iii) x² - 6x – 16 = 0Simultaneous Equations(i) 2x + y = 8 3x – y = 2
(ii) 4x – y = -9
2x – 3y = -7
Other types of equations
(
i
) 3x² - 9x = 0
(ii) x² + 15x = 0
(i) x² - 25 = 0(ii) 4x² -
49 = 0dmccmaths.weebly.com
Taking out what’s common
Difference of 2 squares
28
Mr. D. McCarthy