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