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數 Quadratic Equations 數

Contents 錄 : Quadratic Equations Distinction between identities and equations. Linear equation in one unknown 3 ways to solve quadratic equations 3 Equations transformed to quadratic equations Applications of quadratic equations Simultaneous equations in unknowns 聯立 www5.brinkster.com/andy35 Quadratic Equations 1

(A) Distinction between identities and equations. i. Definition of Equation / Identity / Equation : an equality ( ) which is ture for special value(s) of x ( x). Identity : an equality ( ) which is true for all values of x ( x). ii. Useful Identities : a. b. a + ab + b ( a + b) a ab + b ( a b c. a b ( a + b)( a b) d. 3 3 a + b ( a + b)( a ab + b ) e. 3 3 a b ( a b)( a + ab + b ) f. 4 4 a b ( a + b )( a + b)( a b) ) 黎 Sir : www5.brinkster.com/andy35 Quadratic Equations

Exam Type Questions: a. Determine Identities & Equations e.g. Which is Equation? Which is Identity??? a. x + x + 1 = 0 b. x = 0 c. x + x + 1 = ( x + 1) d. a b = ( a + b)( a b) b. Find Unknowns in identities 數 Skill 1: Compare Coefficient 1: 數 e.g. Find k if ( x 3)( x 1) x 4x + k. ( x 1)( x + ) e.g. For what value(s) of x does the equality ( ) = x + x 1 hold ( 立 )? y + 1 R Sy e.g. Given the identity + ( y 1) 1 y (1 y) (1 y). Find the value of R and S. Skill : Substitution : e.g. Find A, B and C if A ( x + 1)( x ) + B( x 1) + C 3x + x 1 www5.brinkster.com/andy35 Quadratic Equations 3

(B) Linear equation in one unknown ax + b = 0 Linear equation in one unknown e.g. 3x + 7 = 6 e.g. log x + log 5 = log 7 (C) 3 ways to solve quadratic equations 3 Exam Type Questions: a. Solving Quadratic Equations Skill 1: By Factorization - Common Factor 1: - e.g. Solve ( x 1)( x 4) = ( x 4). ( x 1)( x 4) = ( x 4) ( x 1) = 1 x =??? What s wrong? 黎 Sir : www5.brinkster.com/andy35 Quadratic Equations 4

Skill : By Factorization - Useful Identities : - e.g. 4x + 1x + 9 = 0 e.g. 9x 16 = 0 Skill 3: By Factorization - By Cross Method 3: - 6 1 1 e.g. ( ) = x + 1 x + 1 6 e.g. x 3x + = 0 Skill 4: By Formula 4: b ± b 4ac x = where = b 4ac Discriminant a e.g. 4x + 1x + 9 = 0 e.g. 9x 16 = 0 www5.brinkster.com/andy35 Quadratic Equations 5

Skill 5: By Graph 5: Two Steps: 兩 : 1. Plot ( ) the graph of y = ax + bx + c. Find the x-intercept (x ) of y = ax + bx + c, i.e. the roots of ax + bx + c = 0 *** Quadratic Equation : ax + bx + c = 0 ---> Special Case of Quadratic Function 數!!! i.e y = ax + bx + c when y = 0 Quadratic Function: y = ax + bx + c Quadratic Equation: ax + bx + c = 0 e.g. Solve x 3x + = 0. Step 1: Plot ( ) the graph y = x 3x + y y = x 3x + x Step : From the graph, the roots of x 3x + = 0 are 1 and. www5.brinkster.com/andy35 Quadratic Equations 6

b. Nature of Root Skill 1: By Discriminant 1: = b = b = b 4ac > 0, Two Distinct Real Roots 兩不 4ac = 0, Double Real Roots / 4ac < 0, No Real Root y = ax + bx + c y = ax + bx + c y = ax + bx + c = b 4ac > 0 = b 4ac = 0 = b 4ac < 0 e.g. Find the range of p so that 4x + ( p + 3) x = 1 has real roots. p 4x + ( p + 3) x = 1. e.g. Find the value of k if 3x + 4x + k = 0 has equal root. k 3x + 4x + k = 0. e.g. If 9x + kx + 4 = 0 has equal positive roots ( ), find k. k 9x + kx + 4 = 0. www5.brinkster.com/andy35 Quadratic Equations 7

(D) Equations transformed to quadratic equations Exam Type Questions: Skill 1: Fractional Equations 1: 數 1 e.g. + = 1 + x 4x 1 6 8 e.g. = 1 x 1 x + 1 Skill : Equations with degree larger than : e.g. x 4 + x + 1 = 0 4 e.g. x + x 0 = 0 Skill 3: Equations with Surd form 3: e.g. x x = 3 e.g. 4 x 7 x = 0 www5.brinkster.com/andy35 Quadratic Equations 8

Skill 4: 4 Exponential Equations 數 e.g. x x 6 = 0 e.g. 1( 4 x x ) 4( 4 ) = 1 Skill 5: 5 Logarithmic Equations 數 e.g. (log x ) + log x + 1 = 0 e.g. log x log( 3 x) = log 4 (E) Applications of quadratic equations Four Steps to deal with quadratic equations application problems 1. Let the unknown be x 數 x. Set up a quadratic equation in x 立數 x 3. Solve the quadratic equation 4. Answer the question 黎 Sir : www5.brinkster.com/andy35 Quadratic Equations 9

Exam Type Questions: a. Applications of quadratic equations Skill 1: Compound Interests 1: 利 e.g. Andy deposits ( ) $10000 into the saving deposit account ( ) on his 8 th birthday and $9000 half year later. On his 9 th birthday, there is $000 in his account. The annual interest rate ( 年利率 ) for his saving deposit account is x% compounded half-yearly ( 年 ). Find the annual interest rate x%. (Andy 8 了 $10000, 年 $9000. 9, 累了 $000. 年, 年利率 x%. 年利率 x%.) Skill : Length of Rectangle : 度 e.g. Consider a rectangular farm is 864 m. If the length of a farm is increased by 4 m and the width of a farm is reduced ( ) by 3 m, the area ( ) of the new farm is remained the same ( 不 ). Find the original dimension ( 來 ) of the farm. ( 864 m. 度了 4 m 度了 3 m, 不. 來.) www5.brinkster.com/andy35 Quadratic Equations 10

Skill 3: Application of Pythagoras Theorem 3: 理 e.g. A ship leaves port A, sails east to port B, and then north to port C, with a total distance ( 離 ) of 119 km. Tomorrow, the ship sails directly ( ) from port C back to port A with a total distance of 91 km. Find the distance between port A and B. ( 輪 A, B, 北 C, 離 119 km., 輪 C A 離 91 km. A B 離.) Skill 4: Speed Problem 4: 度 e.g. Tom and Jimmy is driving their car from his school to Mary s home, the distance 1 ( 離 ) is 11 km. Jimmy start to drive his car after 3 hours than Tom, but his speed 3 ( 度 ) is 10 km/h faster than Tom s speed. If they arrive at Mary s home at the same time, find the speed of Tom and Jimmy. 1 (Tom Jimmy 車 Mary, 離 11 km. Jimmy Tom 3 3, 度 Tom 度 10 km/h. Mary, Tom Jimmy 度,) www5.brinkster.com/andy35 Quadratic Equations 11

(F) Simultaneous equations in unknowns 聯立 y = ax + bx + c y = mx + c or y = m y = m 1 x + c 1 x + c Exam Type Questions: a. Solving Simultaneous Equations 聯立 Skill 1: Substitution 1: e.g. Skill : Elimination : e.g. www5.brinkster.com/andy35 Quadratic Equations 1

Skill 3: Graphical Method 3: Intersection point of the graph = Solutions of simultaneous equations = 聯立 e.g. L1 and L are straight lines ( ) intersecting ( ) at a point on the y-axis (y ). If the equation ( ) of L is x y 1 = 0, find the equation ( ) of L1 b. Application Problems Skill 1: Price/Quantity Calculation of Goods 1: 兩 / 數量 e.g. The prices ( ) of a rubber and a ruler are $ and $3 respectively ( ). If Andy use totally ( ) $46 to buy some rubbers and rulers for his students. The total number of rubber and rulers is 0. Find the number of rubbers and rulers. ( $ $3. Andy 了 $46, 數 0. 數量.) www5.brinkster.com/andy35 Quadratic Equations 13

Skill : Length/Area of Rectangle : 度 / e.g. The area ( ) of the rectangle is 60 cm. The width ( 度 ) and the length ( 度 ) of a rectangle are x cm and y cm respectively ( ). The length is 1 cm longer than ( ) half of its width ( 度 ). Find the length and width of the rectangle. ( 60 cm. 度 x cm, 度 y cm. 度度 1 cm. 度度.) Skill 3: Double-digit number 3: 數 e.g. A two-digit ( ) student number is increased ( 了 ) by 18 when its digits are reverved ( 數 ). If the sum of the squares of the digits ( 數 ) is 130, find the original student number ( 來數 ). ( 數數, 數了 18. 數 130, 來數.) www5.brinkster.com/andy35 Quadratic Equations 14