Quadratic equations - Wiley

1 OverviewWhy learn this?The Guggenheim Museum in Bilbao (Spain) is covered with thin metal plates like the scales of a fi sh, each one designed and shaped by a computer. This project required the solving of thousands of non linear equations . Parabolic shapes are widely used by engineers and do you know? 1 THInK List what you know about Quadratic equations . Use a thinking tool such as a concept map to show your PaIr Share what you know with a partner and then with a small SHare As a class, create a thinking tool such as a large concept map that shows your class s knowledge of Quadratic sequence8 .1 solving Quadratic equations The Quadratic solving Quadratic equations The Review ONLINE ONLYQ uadratic equationsTOPIC 8number and 30419/08/14 10:03 AMUNCORRECTED PAGE PROOFSWaTCH THIS VIdeOThe story of mathematicsSearchlight Id: 30519/08/14 10:03 AMUNCORRECTED PAGE PROOFS number and algebra306 Maths Quest 10 + solving Quadratic equations algebraicallyQuadratic equations The general form of a Quadratic equation is ax2+bx+c=0.

2 To solve an equation means to fi nd the value of the pronumeral(s) or variables, which when substituted, will make the equation a true Null Factor Law The Null Factor Law states that if the product of two numbers is zero then one or both of the numbers must equal zero. In other words, there are two solutions to the equation pq=0; they are p=0, and q=0. When solving Quadratic equations by applying the Null Factor Law, it is best to write the equations equal to the equation (x 7) (x+11)= the equation and check that the right hand side equals zero. (The product of the two numbers is zero.)(x 7) (x+11)=02 The left hand side is factorised, so apply the Null Factor 7=0 or x+11=03 Solve for x= 11 WOrKed eXamPle 1 WOrKed eXamPle 1 WOrKed eXamPle 1 Solve each of the following x2 3x=0b 3x2 27=0c x2 13x+42=0d 36x2 21x=2 THInKWrITea1 Write the equation.

3 Check that the right hand side equals by taking out the common factor of x2 and 3x, which is (x 3)=03 Apply the Null Factor or x 3=04 Solve for x=3b1 Write the equation. Check that the right hand side equals 27=02 Factorise by taking out the common factor of 3x2 and 27, which is (x2 9)=0x2 3x=0 WOrKed eXamPle 2 WOrKed eXamPle 2 WOrKed eXamPle 30619/08/14 10:03 AMUNCORRECTED PAGE PROOFS number and algebraTopic 8 Quadratic equations 307 solving Quadratic equations by completing the square Sometimes it is necessary to complete the square in order to factorise a Quadratic trinomial. This is often necessary if the solutions are not rational using the difference of two squares (x2 32)=03(x+3) (x 3)=04 Apply the Null Factor +3=0 or x 3=05 Solve for x.

4 X= 3 x=3(Alternatively, x= 3.)c1 Write the equation. Check that the right hand side equals 13x+42=02 Factorise by finding a factor pair of 42 that adds to 13.(x 6) (x 7)=03 Use the Null Factor Law to write two linear 6=0 or x 7=04 Solve for x=7d1 Write the equation. Check that the right hand side equals zero. (It does not.)d36x2 21x=22 Rearrange the equation so the right hand side of the equation equals 21x 2=03 Recognise that the expression to factorise is a Quadratic 24x+3x 2=04 Factorise the (3x 2)+(3x 2)=0(3x 2) (12x+1)=05 Use the Null Factor Law to write two linear 2=0 or 12x+1=03x=212x= 16 Solve for x= 112 Factors of 42 Sum of factors 6 and 7 13 Factors of 72 Sum of factors3 and 24 30719/08/14 10:03 AMUNCORRECTED PAGE PROOFS number and algebra308 Maths Quest 10 + 10 ASolving problems There are many problems that can be modelled by a Quadratic equation.

5 You should fi rst form the Quadratic equation that represents the situation before attempting to solve such problems. Recall that worded problems should always be answered with a the solutions to the equation x2+2x 4=0. Give exact the +2x 4=02 Identify the coeffi cient of x, halve it and square the 2R23 Add the result of step 2 to the equation, placing it after the x term. To balance the equation, we need to subtract the same amount as we have +2x+Q12 2R2 4 Q12 2R2=0x2+2x+(1)2 4 (1)2=0x2+2x+1 4 1=04 Insert brackets around the fi rst three terms to group them and then simplify the remaining terms.(x2+2x+1) 5=05 Factorise the fi rst three terms to produce a perfect square.(x+1)2 5=06 Express as the difference of two squares and then factorise.

6 (x+1)2 (!5)2=0(x+1+!5) (x+1 !5)=07 Apply the Null Factor Law to fi nd linear +1+!5=0 or x+1 !5=08 Solve for x. Keep the answer in surd form to provide an exact 1 !5 or x= 1+!5(Alternatively, x= 1 !5.)WOrKed eXamPle 3 WOrKed eXamPle 3 WOrKed eXamPle 3 When two consecutive numbers are multiplied together, the result is 20. Determine the ne the unknowns. First number = x, second number = x + the two numbers be x and (x+1).2 Write an equation using the information given in the (x+1)=203 Transpose the equation so that the right hand side equals (x+1) 20=04 Expand to remove the +x 20=05 Factorise.(x+5) (x 4)=0 WOrKed eXamPle 4 WOrKed eXamPle 4 WOrKed eXamPle 30819/08/14 10:03 AMUNCORRECTED PAGE PROOFS number and algebraTopic 8 Quadratic equations 309 The height of a football after being kicked is determined by the formula h= +3d, where d is the horizontal distance from the How far away is the ball from the player when it hits the ground?

7 B What horizontal distance has the ball travelled when it first reaches a height of 20 m?THInKWrITea1 Write the d2+3d2 The ball hits the ground when h=0. Substitute h=0 into the formula. d2+3d=03 Factorise. +3d=0d( +3)=04 Apply the Null Factor Law and or +3=0 3 d= 3 the is the origin of the is the distance from the origin that the ball has travelled when it the question in a ball is 30 m from the player when it hits the height of the ball is 20 m, so, substitute h=20 into the +3d20= +3d2 Transpose the equation so that zero is on the right-hand 3d+20=03 Multiply both sides of the equation by 10 to remove the decimal from the coeffi 30d+200=0 WOrKed eXamPle 5 WOrKed eXamPle 5 WOrKed eXamPle 56 Apply the Null Factor Law to solve for x.

8 X+5=0 or x 4=0x= 5x=47 Use the answer to determine the second x= 5, x+1= 4. If x=4, x+1= the :4 5=20 5 4=209 Answer the question in a numbers are 4 and 5 or 5 and 4. 30919/08/14 10:03 AMUNCORRECTED PAGE PROOFS number and algebra310 Maths Quest 10 + 10 AExercise solving Quadratic equations algebraically IndIVIdual PaTHWaYS PraCTISeQuestions:1a f, 2a c, 3a d, 4a c, 5a h, 6 8, 10, 14, 16, 19, 20, 22 COnSOlIdaTeQuestions:1b g, 2a d, 3a f, 4a f, 5d l, 6, 7, 8a d, 9, 11, 14 16, 18, 20, 21, 22, 26 maSTerQuestions:1, 2, 3f i, 4f l, 5g l, 6, 7, 8g l, 9f i, 10d i, 11d i, 12 27 FluenCY1 WE1 Solve each of the following (x+7) (x 9)=0b (x 3) (x+2)=0c (x 2) (x 3)=0 d x(x 3)=0e x(x 1)=0f x(x+5)=0 g 2x(x 3)=0h 9x(x+2)=0i (x 12) (x+12)=0 j (x+ ) (x+ )=0k 2(x ) (2x )=0l (x+!)

9 2) (x !3)=0 2 Solve each of the following (2x 1) (x 1)=0b (3x+2) (x+2)=0c (4x 1) (x 7)=0 d (7x+6) (2x 3)=0e (5x 3) (3x 2)=0f (8x+5) (3x 2)=0 g x(x 3) (2x 1)=0h x(2x 1) (5x+2)=0i x(x+3) (5x 2)=0 3 WE2a Solve each of the following x2 2x=0b x2+5x=0c x2=7xd 3x2= 2xe 4x2 6x=0f 6x2 2x=0 g 4x2 2!7x=0h 3x2+!3x=0i 15x 12x2=0 4 WE2b Solve each of the following x2 4=0b x2 25=0c 3x2 12=0d 4x2 196=0e 9x2 16=0f 4x2 25=0 g 9x2=4h 36x2=9i x2 125=0j 136x2 49=0k x2 5=0l 9x2 11=0 5 WE2c Solve each of the following x2 x 6=0b x2+6x+8=0c x2 6x 7=0d x2 8x+15=0e x2 2x+1=0f x2 3x 4=0reFleCTIOnWhat does the Null Factor Law mean? Individual pathway interactivity int-4601 doc 5256doc 5256doc 5256doc 5257doc 5257doc 5257doc 5257doc 5257doc 5257doc 5258doc 5258doc 5258doc 5258doc 5258doc 5258doc 5259doc 5259doc 5259doc 5260doc 5260doc 52604 Factorise.

10 (d 20) (d 10)=05 Apply the Null Factor 20=0 or d 10= d=107 Interpret the solution. The ball reaches a height of 20 m on the way up and on the way down. The fi r s t time the ball reaches a height of 20 m is the smaller value of d. Answer in a ball fi rst reaches a height of 20 m after it has travelled a distance of 10 31019/08/14 10:04 AMUNCORRECTED PAGE PROOFS number and algebraTopic 8 Quadratic equations 311g x2 10x+25=0h x2 3x 10=0i x2 8x+12=0j x2 4x 21=0k x2 x 30=0l x2 7x+12=0m x2 8x+16=0n x2+10x+25=0o x2 20x+100=06 MC The solutions to the equation x2+9x 10=0 are:a x=1 and x=10b x=1 and x= 10C x= 1 and x=10d x= 1 and x= 10e x=1 and x=9 7 MC The solutions to the equation x2 100=0 are:a x=0 and x=10b x=0 and x= 10C x= 10 and x=10d x=0 and x=100e x= 100 and x=1008 WE2d Solve each of the following 2x2 5x=3b 3x2+x 2=0c 5x2+9x=2d 6x2 11x+3=0e 14x2 11 x=3f 12x2 7x+1=0g 6x2 7x=20h 12x2+37x+28=0 i 10x2 x=2j 6x2 25x+24=0k 30x2+7x 2=0l 3x2 21x= 36 9 WE3 Find the solutions for each of the following equations .