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“Teach A Level Maths” Vol. 1: AS Core Modules 6: Roots, Surds and Discriminant © Christine Crisp Roots, Surds and Discriminant Module C1 "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages" Roots, Surds and Discriminant Roots of Equations Roots is just another word for solutions ! 2 e.g. Find the roots of the equation x 2 x 1 0 Solution: There are no factors, so we can either complete the square or use the quadratic formula. 2 Completing the square: ( x 1) 1 1 0 ( x 1) 2 x 1 2 Using the formula: b b 4ac x 2a 2 x 2 (1) ( 2) 2 4(1)(1) 2 2 8 (2) x 2 Let’s see why the answers (1) and (2) are the same Roots, Surds and Discriminant The answers from the quadratic formula can be simplified: 2 8 We have x 2 Numbers such as 8 are called surds However, 8 4 2 4 is a perfect square so can be square-rooted, so 4 2 2 2 We have simplified the surd 2 8 22 2 So, x x 2 2 2 is a common factor of the numerator, so 1 2(1 2 ) x x 1 2 21 Roots, Surds and Discriminant Exercise Simplify the following surds: 1) 12 4 3 2 3 2) 32 16 2 4 2 3) 45 9 5 3 5 4) 1000 100 10 10 10 Roots, Surds and Discriminant The Discriminant of a Quadratic Function The formula for solving a quadratic equation is b b 2 4ac x 2a The part b 4ac is called the discriminant 2 Because we square root the discriminant, we get different types of roots depending on its sign. Roots, Surds and Discriminant The Discriminant of a Quadratic Function To investigate the roots of the equation x 2 5x 4 0 we consider the graph of the function y x 2 5 x 4 The roots of the equation are at the points where y = 0 ( x = 1 and x = 4) The discriminant y x2 5x 4 b 2 4ac 25 16 9 0 The roots are real and distinct. ( different ) Roots, Surds and Discriminant The Discriminant of a Quadratic Function For the equation x 2 4 x 4 0 the discriminant b 2 4ac 16 16 0 y x2 4x 4 The roots are real and equal ( x = 2) Roots, Surds and Discriminant The Discriminant of a Quadratic Function For the equation x 2 4 x 7 0 . . . . . . the discriminant b 2 4ac 16 28 12 0 y x 2 4x 7 There are no real roots as the function is never equal to zero If we try to solve x 2 4 x 7 0 , we get 4 12 x 2 The square of any real number is positive so there are no real solutions to 12 Roots, Surds and Discriminant SUMMARY The formula for solving the quadratic equation ax bx c 0 2 is b b 2 4ac x 2a The part b 2 4ac is called the discriminant b 2 4ac 0 The roots are real and distinct b 2 4ac 0 ( different ) The roots are real and equal b 2 4ac 0 The roots are not real If we try to solve an equation with no real roots, we will be faced with the square root of a negative number! Roots, Surds and Discriminant Exercise 1 (a) Use the discriminant to determine the nature of the roots of the following quadratic equations: (i) x 2 2 x 1 0 (ii) x 2x 1 0 2 (b) Check your answers by completing the square to find the vertex of the function and sketching. Solution: (a) (i) b 2 4ac 4 4(1)(1) 0 The roots are real and equal. (ii) b 2 4ac 4 4(1)( 1) 4 4 8 The roots are real and distinct. Roots, Surds and Discriminant (b) Check your answers by completing the square to find the vertex of the function and sketching. (b) (i) x 2 2x 1 (x 1) 2 1 1 ( x 1) 2 y x2 2x 1 Vertex is ( -1,0 ) Roots of equation (real and equal) (ii) x 2 2 x2 1 (x 1) 1 1 (x 1) 2 2 Vertex is ( 1,-2 ) Roots of equation (real and distinct) y x2 2x 1 Roots, Surds and Discriminant 2. Determine the nature of the roots of the following quadratic equations ( real and distinct or real and equal or not real ) by using the discriminant. DON’T solve the equations. (a) x 2 6x 9 0 b 2 4ac (6) 2 4(1)(9) 36 36 0 Roots are real and equal (b) 2x 2 5x 9 0 b 2 4ac (5) 2 4(2)(9) 25 72 47 0 There are no real roots (c) 5x 2 9x 2 0 b 2 4ac (9) 2 4(5)( 2) 81 40 121 0 Roots are real and distinct Roots, Surds and Discriminant Roots, Surds and Discriminant The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet. Roots, Surds and Discriminant Roots is just another word for solutions ! 2 x 2x 1 0 e.g. Find the roots of the equation Solution: There are no factors, so we can either complete the square or use the quadratic formula. 2 Completing the square: ( x 1) 1 1 0 ( x 1) 2 x 1 2 b b 4ac x 2a 2 2 2 4(1)( 1) x 2 2 8 x 2 Using the formula: 2 (1) (2) Roots, Surds and Discriminant The answers from the quadratic formula can be simplified: 2 8 We have x 2 Numbers such as 8 are called surds However, 8 4 2 4 is a perfect square so can be square-rooted, so 4 2 2 2 We have simplified the surd 2 8 22 2 x So, x 2 2 2 is a common factor of the numerator, so 1 2(1 2 ) x x 1 2 21 Roots, Surds and Discriminant The Discriminant The formula for solving the quadratic equation 2 b b 4ac 2 x ax bx c 0 is 2a The part b 2 4ac is called the discriminant b 2 4ac 0 The roots are real and distinct. ( different ) The roots are real and equal. b 2 4ac 0 The roots are not real. b 2 4ac 0 If we try to solve an equation with no real roots, we will be faced with the square root of a negative number! Roots, Surds and Discriminant e.g. For y x2 5x 4 The discriminant b 2 4ac 25 16 9 0 y x2 5x 4 The roots are real and distinct. ( different ) Roots, Surds and Discriminant e.g. For y x 2 4 x 4 The discriminant b 2 4ac 16 16 0 y x2 4x 4 The roots are real and equal. Roots, Surds and Discriminant e.g. For y x 2 4 x 7 The discriminant b 2 4ac 16 28 12 0 y x2 4x 7 There are no real roots as the function is never equal to zero.