VS 2024 EM3 #1 Quadratic and Fractional Equations (Solution)

1 Name: ____Teacher’s Solution_____ Date: _____________________ Class 3 Reg No CHAPTER 1: QUADRATIC AND FRACTIONAL EQUATIONS Reference Book: think! Mathematics Secondary Textbook, Shinglee In this chapter you will learn how to: • Solve quadratic equations in one unknown variable by o Factorisation o Completing the square for 2x px q++ o Use of formula 2 4 2 b b acx a −± −= where the quadratic equation is of the form 2 0ax bx c+ += and a, b and c are real constants o Graphical Method (we will learn this later) • Solve fractional equations that can be reduced to quadratic equations • Formulate a quadratic equation in one variable to solve problems • Sketch the graphs of quadratic equations of the form ( )( ) ,y xhxk= −− ( )( ) ,y xhxk= −− − ( ) 2 y xp q= −+ and ( ) 2 .y xp q= −− + 1 Introduction A quadratic expression is in the form of 2ax bx c++ , where ,,abc are real numbers, and 0a≠ . 1.1 Solving Quadratic Equations by Factorisation (Recap) Last year, we learnt how to solve a quadratic equation by using the Zero Product Property: If 0, then either 0 or 0.AB A B×= = = This property can be extended to product of several terms, i.e. If 0, then either 0 or 0 or 0 or 0ABCD A B C D×××= = = = = We thus need to express a quadratic equation in the form where RHS is zero, i.e. 2 0ax bx c+ += The quadratic on the LHS will need to be changed into product of factors by factorising highest common factor, or using grouping, algebraic identities or ‘cross’ method. VICTORIA SCHOOL S3 ELEMENTARY MATHEMATICS

2 Class Practice 1 1 Solve each of the following equations. (a) 35 125 0aa−= (b) ( )312−=xx (c) (7 3 )( 2) 4xx− += (d) 32 10yyy+ ++= Note: To solve an equation involving an unknown means to find the values of the unknown that will make the equation true. The values are referred to as solutions to the equations or roots of the equation. If a and b are roots of a quadratic equation, then the factors of it are ()xa− and ()xb− . ( ) ( )( ) 3 22 5 125 0 5 50 5 5 50 0 or 5 or 5 aa aa aa a aa a −= −= + −= == −= ( ) ( )( ) 2 312 3 20 32 10 3 2 0 or 1 0 2 13 xx xx xx xx xx −= −−= + −= += −= =−= ( )( ) 2 2 2 (7 3 )( 2) 4 7 14 3 6 4 3 14 4 3 10 0 35 20 3 5 0 or 2 0 5 23 xx x xx xx xx xx xx xx − += +− −= − ++ = −− = + −= += −= =−= ( ) ( ) ( ) ( ) 32 2 2 2 2 2 10 1 10 1 10 10 o r 10 11 (reject as is always positive) yyy yy y yy yy yy y + ++= ++ += + += += += =− =−

3 Example Form a quadratic equation in x with the given roots (a) 3, 4− (b) 24,35 − 1.2 Solving Quadratic Equations by Completing the Square 1.2.1 Solving Quadratic Equations of the f orm ( ) 2 x+a =b Example Solve each of the following equations. (a) ( ) 2 5 4 81−=x (b) 2 1 102  −= x 1.2.2 Completing the square for quadratic expression of the form +2x px When direct factorisation cannot be applied, it will be useful to rewrite the equation into the form 2()xp q+= where p and q are real numbers. Recall your identity: 22 2() 2a b a ab b+= ++ In particular, 22 2() 2xa x a xa+= ++ or 22 2() 2x a x ax a−= −+ Note: () + 2xa is considered a perfect square.

4 Let us consider the expansion of ( ) 2 3.x+ ( ) 2 3x+= Try to arrange this into a square. Recall that, ( ) ( ) ( ) 2 3 3 3.x xx+ =+×+ In general, quadratic equations of the form ( ) 2 xa+ can be arranged into a multiplication frame similar to what we did above. What about 2 6?xx+ × x 3 x 3 What is the number that we must add to complete the square? __9____ However, ( ) 22 6 3.x xx+≠+ Since we need to add ____ to 2 6,xx+ we must _________________ so that the original expression does not change. 2 6xx+= From this we realised that to make a quadratic expression of the form 2x px+ into a perfect square ( ) 2 xa+ we have to add a number, b. × x 3 x 3 × x 3 x 3 × x 3 x 3

5 × x 3 x 3 × × × Let us find out what is the number that we have to add. Quadratic expression 2x px+ Number that must be added to complete the square, b 2 coefficient of 2 px = Quadratic expression of the form ( ) 2 xa b+− (a) 2 6xx+ 239= 6 2 = 3 ( ) 2 2 22 2 6 6 33 39 xx xx x + =++− = +− (b) 2 4xx+ (c) 2 8xx+ (d) 2 10xx+ 1. What is the relationship between b and p? ___________ 2. What is the relationship between a and p? ___________ H ence, to make 𝑥𝑥2 + 𝑝𝑝 𝑥𝑥 a perfect square, 22x px x px+=++ 2 2 x px x +=+ −

6 In general, to make 2x px+ into a perfect square, ________must be added to it, where p is the coefficient of x. Example Complete the following expressions such that they are perfect squares. Complete the following expressions such that they are perfect squares. Express 2x px+ in completed square form. (1) 2 2 77 2 ++  xx = 2 7 2x + 22 2 777 22x xx   +=+ −     (2) 2 2 33 2xx −+  = 2 3 2x − 22 2 333 22x xx   −=− −     (3) 2 2 77 24xx ++  = 2 7 4x + 22 2 7 77 2 44x xx   += +−     (4) 2 2 14 7 33cc ++  = 2 7 3c + 22 2 14 7 7 3 33c cc   += +−      (5) 2 2 77 8 16vv −+  = 2 7 16v + 22 2 7 77 8 16 16v vv  −= + −   Note that coefficient of 2x ___________________. 1.2.3 Completing the square for quadratic expression of the form ++2x px q If instead of 2x px+ , we are given the expression 2x px q++ , we can still convert the first two terms into perfect squares. Example Express the following in the form ( ) 2 xa c++ , where a and c are real numbers. (a) 2 6 11xx++ (b) 22 83xx−+ ( ) ( ) 2 22 2 2 6 11 66 1122 3 9 11 32 xx x x x ++   =+− +     = + −+ = ++ ( ) ( ) ( ) ( ) 2 2 22 2 2 2 2 83 2 43 4423 22 2 2 43 2 2 83 2 25 xx x x x x x −+ = −+   = −− +     = − −+ = − −+ = −−

7 In conclusion, 1.2.4 Solving E quations of the form + +=2 0x px q by using Completing the Square Steps to solve a quadratic equation by completing the square [add − q to both sides of the equation] [complete the square on LHS] 22 2 22 ppx px x   +=+ −     ( ) 22 22 22 x px q x px q ppxq + += + +  = +− +   [Make the perfect square term the subject] [Take square root on both sides. Remember to add plus-minus sign] [Make x the subject]

8 Example Solve 2 6 60xx+ += . 2 6 60xx+ += Class Practice 2 1 Express each of the following expressions in the form 2( ).xa b++ (a) 2 91+−xx (b) 2 1.4−xx 2 Solve each of the following equations. (a) 2 3 40xx+ −= (b) 22 0.6 1 0xx+ −= ( ) ( ) 2 2 2 66 396 33 33 33 4.73 or 1.27 xx x x x x xx += − + −= − += += ± = −± ≈− ≈− 2 22 2 2 91 99 122 9 81 124 9 85 24 xx x x x +−   =+− −     =+ −− = +− ( ) ( ) ( ) 2 22 22 2 1.4 1.4 1.4 22 0.7 0.7 0.7 0.49 xx x x x −  = −−   = −− = −− 2 2 22 2 2 3 40 34 33 422 39 424 3 25 24 3 25 24 35 22 4 or 1 It is much faster to factorise! xx xx x x x x x xx + −= +=   +− =       +=+      +=  += ± = −± = −= ( ) ( ) ( ) ( ) 2 2 2 22 22 2 2 0.6 1 0 0.3 0.5 0 0.