Algebraic Operations Summary of Factorising Methods

Created by Mr. Lafferty@mathsrevision.com
Algebraic Operations Summary of Factorising Methods Introduction to Quadratic Equation Factorising Trinomials (Quadratics) Real-life Problems on Quadratics Finding roots by factorising and formula Exam Type Questions 14-Nov-18 Created by Mr.

Starter Questions S4 Credit Q1. Remove the brackets (a) y(4y – 3x) = (b) (x + 5)(x - 5) = Q2. For the line y = -x + 5, find the gradient and where it cuts the y axis. Q3. Find the highest common factor for p2q and pq2. 14-Nov-18 Created by Mr.

Created by Mr. Lafferty@www.mathsrevision.com
Factorising Methods S4 Credit Learning Intention Success Criteria To review the three basic methods for factorising. To be able to identify the three methods of factorising. Apply knowledge to problems. 14-Nov-18 Created by Mr.

Summary of Factorising S4 Credit When we are asked to factorise there is priority we must do it in. Take any common factors out and put them outside the brackets. 2. Check for the difference of two squares. 3. Factorise any quadratic expression left. 14-Nov-18 Created by Mr.

Common Factor S4 Credit Factorise the following : 2x(y – 1) (a) 4xy – 2x (b) y2 - y y(y – 1) 14-Nov-18 Created by Mr.

a2 – b2 Difference of Two Squares
S4 Credit When we have the special case that an expression is made up of the difference of two squares then it is simple to factorise The format for the difference of two squares a2 – b2 First square term Second square term Difference 14-Nov-18 Created by Mr.

a2 – b2 ( a + b )( a – b ) Difference of Two Squares
Check by multiplying out the bracket to get back to where you started S4 Credit a2 – b2 First square term Second square term Difference This factorises to ( a + b )( a – b ) Two brackets the same except for + and a - 14-Nov-18 Created by Mr.

Always the difference sign -
Difference of Two Squares S4 Credit Keypoints Format a2 – b2 Always the difference sign - ( a + b )( a – b ) 14-Nov-18 Created by Mr. Lafferty

Difference of Two Squares ( w + z )( w – z ) ( 3a + b )( 3a – b )
S4 Credit Factorise using the difference of two squares ( w + z )( w – z ) (a) w2 – z2 (b) 9a2 – b2 (c) 16y2 – 100k2 ( 3a + b )( 3a – b ) ( 4y + 10k )( 4y – 10k ) 14-Nov-18 Created by Mr. Lafferty

Difference of Two Squares 6(x + 2 )( x – 2 ) 3( w + 1 )( w – 1 )
S4 Credit Factorise these trickier expressions. 6(x + 2 )( x – 2 ) (a) 6x2 – 24 3w2 – 3 8 – 2b2 (d) 27w2 – 12 3( w + 1 )( w – 1 ) 2( 2 + b )( 2 – b ) 3(3 w + 2 )( 3w – 2 ) 14-Nov-18 Created by Mr. Lafferty

Factorising x2 + 3x + 2 ( ) ( ) x x + 2 + 2 x x + 1 + 1
Using St. Andrew’s Cross method Strategy for factorising quadratics Find two numbers that multiply to give last number (+2) and Diagonals sum to give middle value +3x. x2 + 3x + 2 x x + 2 + 2 (+2) x( +1) = +2 x x + 1 + 1 (+2x) +( +1x) = +3x ( ) ( ) 14-Nov-18 Created by Mr.

Factorising x2 + 6x + 5 ( ) ( ) x x + 5 + 5 x x + 1 + 1
Using St. Andrew’s Cross method Strategy for factorising quadratics Find two numbers that multiply to give last number (+5) and Diagonals sum to give middle value +6x. x2 + 6x + 5 x x + 5 + 5 (+5) x( +1) = +5 x x + 1 + 1 (+5x) +( +1x) = +6x ( ) ( ) 14-Nov-18 Created by Mr.

Factorising x2 - 4x + 4 ( ) ( ) x x - 2 - 2 x x - 2 - 2
Both numbers must be - Factorising Using St. Andrew’s Cross method Strategy for factorising quadratics Find two numbers that multiply to give last number (+4) and Diagonals sum to give middle value -4x. x2 - 4x + 4 x x - 2 - 2 (-2) x( -2) = +4 x x - 2 - 2 (-2x) +( -2x) = -4x ( ) ( ) 14-Nov-18 Created by Mr.

Factorising x2 - 2x - 3 ( ) ( ) x x - 3 - 3 x x + 1 + 1
One number must be + and one - Factorising Using St. Andrew’s Cross method Strategy for factorising quadratics Find two numbers that multiply to give last number (-3) and Diagonals sum to give middle value -2x x2 - 2x - 3 x x - 3 - 3 (-3) x( +1) = -3 x x + 1 + 1 (-3x) +( x) = -2x ( ) ( ) 14-Nov-18 Created by Mr.

Using St. Andrew’s Cross method
Factorising Using St. Andrew’s Cross method S4 Credit Factorise using SAC method (m + 1 )( m + 1 ) (a) m2 + 2m +1 y2 + 6m + 5 b2 – b -2 (d) a2 – 5a + 6 ( y + 5 )( y + 1 ) ( b - 2 )( b + 1 ) ( a - 3 )( a – 2 ) 14-Nov-18 Created by Mr. Lafferty

Factorising Methods S4 Credit Now try MIA Ex 1.1 Ch8 (page156) 14-Nov-18 Created by Mr.

Starter Questions Q1. True or false y ( y + 6 ) -7y = y2 -7y + 6
S4 Credit Q1. True or false y ( y + 6 ) -7y = y2 -7y + 6 Q2. Fill in the ? 49 – 4x2 = ( ? + ?x)(? – 2?) Q3. Write in scientific notation 14-Nov-18 Created by Mr.

Quadratic Equations f(x) = a x2 + b x + c www.mathsrevision.com
This is called a quadratic equation Quadratic Equations S4 Credit A quadratic function has the form a , b and c are constants and a ≠ 0 f(x) = a x2 + b x + c The graph of a quadratic function has the basic shape y The x-coordinates where the graph cuts the x – axis are called the Roots of the function. y x x i.e. a x2 + b x + c = 0

The graph of a quadratic function is called a parabola
Quadratic Equations This is the graph of a golf shot. h The height h m of the ball after t seconds is given by : h = 15t – 5t2 The graph of a quadratic function is called a parabola (a) For what values t does h = 0 t (b) What are the solutions for t = 0 t = 3 15t – 5t2 = 0 t = 0 and t = 3

(b) What is the value of h for
Quadratic Equations h This is the graph of a parabola h = 10t – 2t2 (a) From the graph, what are the roots of the quadratic eqn. Both 8 h = 10t – 2t2 (b) What is the value of h for t = 1 and t = 4 t t = 0 t = 5 (c) What are the solutions of the quadratic equation 10t – 2t2 = 0 t = 0 and t = 5 (d) What is the solution of the quadratic equation 10t – 2t2 = 12.5 2.5

Quadratic Equation S4 Credit Now try MIA Ex2.1 Q2 & Q4 Ch8 (page 158) 14-Nov-18 Created by Mr.

Starter Questions Q1. Multiple out the brackets and simplify.
S4 Credit Q1. Multiple out the brackets and simplify. (a) ( 2x – 5 )( x + 5 ) Q2. Find the volume of a cylinder with height 6m and diameter 9cm Q3. True or false the gradient of the line is 1 x = y + 1 14-Nov-18 Created by Mr.

Solving Quadratic Equations
Factors and Solving Quadratic Equations S4 Credit Learning Intention Success Criteria To explain how factors help to solve quadratic equations. Be able find factors using the three methods to solve quadratic equations. 14-Nov-18 Created by Mr.

Factors and Solving Quadratic Equations S4 Credit The main reason we learn the process of factorising is that it helps to solve (find roots) for quadratic equations. Reminder of Methods Take any common factors out and put them outside the brackets. 2. Check for the difference of two squares. 3. Factorise any quadratic expression left. 14-Nov-18 Created by Mr.

Examples S4 Credit Solve ( find the roots ) for the following x2 – 4x = 0 Common Factor 16t – 6t2 = 0 Common Factor x(x – 4) = 0 2t(8 – 3t) = 0 x = 0 and x - 4 = 0 2t = 0 and 8 – 3t = 0 x = 4 t = 0 and t = 8/3 14-Nov-18 Created by Mr.

Examples S4 Credit Solve ( find the roots ) for the following x2 – 9 = 0 100s2 – 25 = 0 Difference 2 squares Difference 2 squares (10s – 5)(10s + 5) = 0 (x – 3)(x + 3) = 0 10s – 5 = 0 and 10s + 5 = 0 x = 3 and x = -3 s = 0.5 and s = - 0.5 14-Nov-18 Created by Mr.

Now try MIA Ex 3.1 Ch8 (page 159) Factors and
Solving Quadratic Equations S4 Credit Now try MIA Ex 3.1 Ch8 (page 159) 14-Nov-18 Created by Mr.

Examples S4 Credit 2x2 – 8 = 0 Common Factor 80 – 125e2 = 0 Common Factor 2(x2 – 4) = 0 5(16 – 25e2) = 0 Difference 2 squares Difference 2 squares 2(x – 2)(x + 2) = 0 5(4 – 5e)(4 + 5e) = 0 (x – 2)(x + 2) = 0 (4 – 5e)(4 + 5e) = 0 4 – 5e = 0 and 4 + 5t = 0 (x – 2) = 0 and (x + 2) = 0 x = 2 and x = - 2 e = 4/5 and e = - 4/5

Examples S4 Credit Solve ( find the roots ) for the following x2 + 5x + 4 = 0 1 + x - 6x2 = 0 SAC Method SAC Method x 4 1 +3x x 1 1 -2x (x + 4)(x + 1) = 0 (1 + 3x)(1 – 2x) = 0 x + 4 = 0 and x + 1 = 0 1 + 3x = 0 and 1 - 2x = 0 x = - 4 and x = - 1 x = - 1/3 and x = 0.5

Examples S4 Credit Multiply out and rearrange Multiply out and rearrange Solve ( find the roots ) for the following (x + 4)2 =36 5x(2x + 1) - 10 = x(7x + 6) x2 + 8x - 20 = 0 3x2 - x - 10 = 0 SAC Method SAC Method x 10 3x +5 x - 2 x -2 (x + 10)(x - 2) = 0 (3x + 5)(x – 2) = 0 x + 10 = 0 and x - 2 = 0 3x + 5 = 0 and x - 2 = 0 x = - 10 and x = + 2 x = - 5/3 and x = 2

Multiply through by 2(x - 1)(x + 2) to remove denominators Solving Quadratic Equations Examples S4 Credit Solve ( find the roots ) for the following x - 4 x 1 2(x + 2) + 2(x – 1) = (x – 1)(x + 2) 2x x – 2 = x2 + x - 2 (x - 4)(x + 1) = 0 x2 - 3x – 4 = 0 x - 4 = 0 and x + 1 = 0 SAC Method x = 4 and x = - 1

Multiply through by x(x + 1) to remove denominators Solving Quadratic Equations Examples S4 Credit Solve ( find the roots ) for the following x 3 x - 2 6(x + 1) - 6x = x(x + 1) 6x + 6 – 6x = x2 + x (x + 3)(x - 2) = 0 x2 + x – 6 = 0 x + 3 = 0 and x - 2 = 0 SAC Method x = - 3 and x = 2

Starter Questions www.mathsrevision.com S4 Credit
created by Mr. Lafferty

Real-life Quadratics www.mathsrevision.com Learning Intention
S4 Credit Learning Intention Success Criteria To show how quadratic theory is used in real-life. To be able to using quadratic theory in real-life problem. created by Mr. Lafferty

Real-life Problems x2 = 100 x = 10 x = -10
A rectangle garden is twice as long as it is wide. The area is 200m2. Find the dimensions of the rectangle garden. Let width be x Area = length x breadth Length is 2x 200 = 2x x x 200 = 2x2 x2 = 100 x = 10 x = -10 and x must be positive ( We cannot get a negative length !!! ) Width is equal to 10m Length is equal to 20m

Exam Type Questions

Real-life Problems The height in metres of a rocket fired vertically upwards is give by the formula : h = 176t – 16t2 (a) When will the rocket be at a height of 160 metres. 160 = 176t – 16t2 16t t = 0 t2 - 11t + 10 = 0 (t – 10)(t – 1) = 0 t = 10 and t = 1 (b) Is it possible for the rocket to h = 500 metres. Since 500 = 176t -16t2 has no solution not possible.

Real-life Quadratics S4 Credit Now try MIA Ex 5.1 & 5.2 Ch8 (page 164) 14-Nov-18 Created by Mr.

Starter Questions www.mathsrevision.com S4 Credit

Roots Formula www.mathsrevision.com Learning Intention
S4 Credit Learning Intention Success Criteria To explain how to find the roots (solve) quadratic equations by use quadratic formula. To be able to solve quadratic equations using quadratic formula. created by Mr. Lafferty

Every quadratic equation can be rearranged into the standard form
Roots Formula S4 Credit Every quadratic equation can be rearranged into the standard form a, b and c are constants ax2 + bx + c = 0 Examples : find the constants a, b and c for the following 3x2 + x + 4 = 0 a = 3 b = 1 c = 4 x2 - x - 6 = 0 a = 1 b = -1 c = -6 x(x - 2) = 0 x2 – 2x = 0 a = 1 b = -2 c = 0 created by Mr. Lafferty

Roots Formula Now try MIA Ex6.1 First Column (page 166)
S4 Credit Now try MIA Ex6.1 First Column (page 166) created by Mr. Lafferty

Roots Formula ax2 + bx + c = 0 www.mathsrevision.com
S4 Credit Every quadratic equation can be rearranged into the standard form a, b and c are constants ax2 + bx + c = 0 In this form we can use the quadratic root formula to find the roots. created by Mr. Lafferty

Roots Formula ax2 + bx + c = 0 www.mathsrevision.com
S4 Credit Example : Solve x2 + 3x – 3 = 0 ax2 + bx + c = 0 1 3 -3 created by Mr. Lafferty

Roots Formula www.mathsrevision.com and and S4 Credit

Roots Formula Use the quadratic formula to solve the following :
2x2 + 4x + 1 = 0 x2 + 3x – 2 = 0 x = -1.7, -0.3 x = -3.6, 0.6 5x2 - 9x + 3 = 0 3x2 - 3x – 5 = 0 x = 1.4, 0.4 x = 1.9, -0.9 created by Mr. Lafferty

Roots Formula Now try MIA Ex7.1 & 7.2 (page 168) www.mathsrevision.com
S4 Credit Now try MIA Ex7.1 & 7.2 (page 168) created by Mr. Lafferty

Exam Type Questions

Algebraic Operations Summary of Factorising Methods

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