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. Lafferty@mathsrevision.com

Created by Mr. Lafferty@mathsrevision.com 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. www.mathsrevision.com Q3. Find the highest common factor for p2q and pq2. 14-Nov-18 Created by Mr. Lafferty@mathsrevision.com

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. www.mathsrevision.com 14-Nov-18 Created by Mr. Lafferty@www.mathsrevision.com

Created by Mr. Lafferty@www.mathsrevision.com 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. www.mathsrevision.com 3. Factorise any quadratic expression left. 14-Nov-18 Created by Mr. Lafferty@www.mathsrevision.com

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

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 www.mathsrevision.com a2 – b2 First square term Second square term Difference 14-Nov-18 Created by Mr. Lafferty@www.mathsrevision.com

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 www.mathsrevision.com ( a + b )( a – b ) Two brackets the same except for + and a - 14-Nov-18 Created by Mr. Lafferty@www.mathsrevision.com

Always the difference sign - Difference of Two Squares S4 Credit Keypoints Format a2 – b2 www.mathsrevision.com 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 ) www.mathsrevision.com ( 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 ) www.mathsrevision.com 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. Lafferty@mathsrevision.com

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. Lafferty@mathsrevision.com

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. Lafferty@mathsrevision.com

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. Lafferty@mathsrevision.com

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 ) www.mathsrevision.com ( b - 2 )( b + 1 ) ( a - 3 )( a – 2 ) 14-Nov-18 Created by Mr. Lafferty

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

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?) www.mathsrevision.com Q3. Write in scientific notation 0.0341 14-Nov-18 Created by Mr. Lafferty@mathsrevision.com

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 www.mathsrevision.com 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

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

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 www.mathsrevision.com Q3. True or false the gradient of the line is 1 x = y + 1 14-Nov-18 Created by Mr. Lafferty@mathsrevision.com

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. www.mathsrevision.com 14-Nov-18 Created by Mr. Lafferty@www.mathsrevision.com

Solving Quadratic Equations 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. www.mathsrevision.com 2. Check for the difference of two squares. 3. Factorise any quadratic expression left. 14-Nov-18 Created by Mr. Lafferty@www.mathsrevision.com

Solving Quadratic Equations 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 www.mathsrevision.com x = 4 t = 0 and t = 8/3 14-Nov-18 Created by Mr. Lafferty@www.mathsrevision.com

Solving Quadratic Equations 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 www.mathsrevision.com (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. Lafferty@www.mathsrevision.com

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) www.mathsrevision.com 14-Nov-18 Created by Mr. Lafferty@www.mathsrevision.com

Solving Quadratic Equations 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 www.mathsrevision.com 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

Now try MIA Ex 3.2 Ch8 (page 160) Factors and Solving Quadratic Equations S4 Credit Now try MIA Ex 3.2 Ch8 (page 160) www.mathsrevision.com 14-Nov-18 Created by Mr. Lafferty@www.mathsrevision.com

Solving Quadratic Equations 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 www.mathsrevision.com 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

Now try MIA Ex 4.1 Ch8 (page 161) Factors and Solving Quadratic Equations S4 Credit Now try MIA Ex 4.1 Ch8 (page 161) www.mathsrevision.com 14-Nov-18 Created by Mr. Lafferty@www.mathsrevision.com

Solving Quadratic Equations 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 www.mathsrevision.com 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

Solving Quadratic Equations 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) www.mathsrevision.com 2x + 4 + 2x – 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

Solving Quadratic Equations 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) www.mathsrevision.com 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

Now try MIA Ex 4.2 Ch8 (page 162) Factors and Solving Quadratic Equations S4 Credit Now try MIA Ex 4.2 Ch8 (page 162) www.mathsrevision.com 14-Nov-18 Created by Mr. Lafferty@www.mathsrevision.com

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. www.mathsrevision.com 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 16t2 - 176t + 160 = 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.

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

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

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. www.mathsrevision.com 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 www.mathsrevision.com 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) www.mathsrevision.com 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. www.mathsrevision.com 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 www.mathsrevision.com created by Mr. Lafferty

Roots Formula www.mathsrevision.com and and S4 Credit created by Mr. Lafferty

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) www.mathsrevision.com created by Mr. Lafferty

Exam Type Questions

Exam Type Questions

Exam Type Questions

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Exam Type Questions