Quadratic equations. Before you start this chapter. This chapter is about quadratics. Objectives In nature, the growth of a

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1 20 Quadratic equations This chapter is about quadratics. Objectives In nature, the growth of a This chapter will show you how to population of rabbits can be factorise quadratic expressions, including the difference of two squares modelled by a quadratic solve quadratic equations by rearranging equation. factorise quadratics and solve quadratic equations of the form ax 2 1 bx 1 c 5 0 * use the quadratic equations formula * complete the square * efore you start this chapter Put your calculator away! 1 Factorise a 3x 1 6y b 8x 1 x 2 c 3m 2 1 mn d 5r rt e 12xyz 1 6xy 1 18y 2 HELP Chapter 12 2 Work out a b c d (2 1 24) 3 23 e (26) f Work out the value of these expressions when a 5 23, b 5 24 and c 5 2. a abc b 24b HELP Chapter 15 c ab 2 2 c d b 2 2 4ac

2 20.1 L Why learn this? eing able to factorise a quadratic expression will help when solving quadratic equations. Factorising the difference of two squares Objectives Factorise a quadratic expression that is the difference of two squares Keywords quadratic expression, difference of two squares, factorise Skills check HELP Section Expand a (x 1 3)(x 2 3) b (x 1 7)(x 2 7) c (x 2 5)(x 1 5) What do you notice when you have expanded the brackets? 2 Expand a (2a 1 4)(2a 2 4) b (3x 2 2)(3x 1 2) c (5m 1 n)(5m 2 n) Quadratic expressions quadratic expression is an algebraic expression whose highest power of x is x 2. They are usually of the form ax 2 1 bx 1 c, where a, b and c are numbers and a 0. These are all quadratic expressions. 3x 2 1 2x 1 5 x 2 2 3x 2 2 4x x 2 2 3x These expressions all represent one square number subtracted from another. x c a n expression of the form x 2 2 b 2, where x and b are numbers or algebraic terms, is called the difference of two squares. In general, x 2 2 b 2 5 (x 2 b)(x 1 b). Check this by multiplying out (x 2 b)(x 1 b). Example 1 Factorise x Remember that factorising is the inverse of expanding brackets. x x (x 2 3)(x 1 3) Write as letter squared 2 number squared. Use x 2 2 b 2 5 (x 2 b)(x 1 b) with b 5 3. O2 Exercise 20 1 Factorise a x b x c x d x e a f m g n h t Joe thinks of a number, squares it and subtracts 16. a Write down an algebraic expression to illustrate this. b Factorise your answer to part a. 298 Quadratic equations

3 Example 2 Factorise 16m m (4m) Notice that 16m 2 5 (4m) 2. 5 (4m 2 7)(4m 1 7) Exercise 20 1 Factorise a 4x b 9a c 16m d 100t e 169z f 225q Copy and complete. Take out 2 as a common factor. 72m ( 2 ) 52( 2 )( 1 ) 3 Factorise each expression by first taking out a common factor. a 50x b 27m c 80t d 2a b 2 e 3h k 2 f 600x 2 2 6y 2 L 20.2 Factorising quadratics of the form x 2 1 bx 1 c Why learn this? Understanding how an algebraic expression is constructed can tell you much more about the expression. Objectives Factorise a quadratic expression of the form x 2 1 bx 1 c Keywords product, sum, coefficient Skills check 1 Find two positive numbers whose a product is 12 and sum is 7 b product is 20 and sum is 12 c product is 12 and sum is 13 d product is 212 and sum is 1 e product is 212 and sum is 21 f product is 212 and sum is 24. Factorising quadratics of the form x 2 1 bx 1 c Expanding a product of two expressions, like (x 1 2) and (x 1 3), gives a quadratic expression. (x 1 2)(x 1 3) 5 x 2 1 5x Factorising quadratics of the form x 2 1 bx 1 c 299

4 Factorising is the inverse of expanding. To factorise a quadratic, you need to write it as the product of two expressions. 5 is the sum of 2 and 3. x 2 1 5x x 2 1 2x 1 3x (x 1 2)(x 1 3) 6 is the product of 2 and 3. In general, to factorise the equation x 2 1 bx 1 c, find two numbers whose sum is b (the coefficient of x) and whose product is c. The coefficient of x is the number multiplying the x. Example 3 Factorise x 2 1 7x x 2 1 7x (x 1 5)(x 1 2) First look for the product, then test the sums. Find two numbers whose product is 10 and whose sum is 7. The pairs of numbers whose product is 10 are 1 and and The numbers must be 2 and 5. Exercise 20C 1 Factorise each quadratic expression. a x 2 1 5x 1 6 b x 2 1 6x 1 8 c z 2 1 6z 1 5 d a a 1 10 e n 2 1 8n 1 15 f f f 1 36 g m 2 1 8m 1 12 h x x 1 24 i b b 1 30 Example 4 Factorise x 2 2 7x x 2 2 7x (x 2 2)(x 2 5) Exercise 20D Find two numbers whose product is 10 and whose sum is 27. The pairs of numbers whose product is 10 are 1 and and and and The numbers must be 22 and Factorise each quadratic expression. a x 2 2 5x 1 6 b x 2 2 9x 1 8 c z 2 2 7z 1 12 d a 2 2 9a 1 18 e n n 1 25 f f 2 2 8f 1 16 g x x 1 30 h b b 1 28 i p p Quadratic equations

5 Example 5 Factorise a x 2 2 6x 2 7 b x 2 1 x 2 12 a x 2 2 6x (x 2 7)(x 1 1) b x 2 1 x (x 2 3)(x 1 4) Find two numbers whose product is 27 and whose sum is 26. The pairs of numbers whose product is 27 are 21 and and The numbers must be 1 and 27. With practice, you will become better at spotting the correct combination. Find two numbers whose product is 212 and whose sum is 1. The pairs of numbers whose product is 212 are 21 and and and and and and The numbers must be 23 and 4. General rules for factorising quadratics In general if c is positive and b is positive, both numbers in the brackets will be positive if c is positive and b is negative, both numbers in the brackets will be negative if c is negative, one number will be negative, one will be positive. Exercise 20E 1 Factorise a x 2 1 4x 2 12 b x 2 2 x 2 20 c z 2 2 2z 2 15 d a 2 1 6a 2 7 e n 2 1 6n 2 16 f f 2 2 f 2 30 g m 2 1 m 2 30 h t 2 2 6t 2 72 i y y Copy and complete these statements. a t 2 1 7r 2 5 (t 1 10)(t 2 ) b m (t )(t 2 5) c q q 5 (q )(q 2 2) 3 a Factorise each expression. Simplify your answers as much as possible. i x 2 1 6x 1 9 ii x 2 2 8x 1 16 iii x 2 1 4x 1 4 iv x x 1 49 v x x 1 25 vi x x 1 64 b What do you notice about all the answers to part a? c Copy and complete these statements, where m and n are numbers. i (x 1 m) 2 5 x 2 1 x 1 ii (x 2 n) 2 5 x 2 2 x Factorising quadratics of the form x 2 1 bx 1 c 301

6 20.3 Why learn this? The path of a cricket ball can be modelled using a quadratic equation. Solving quadratic equations Objectives Solve quadratic equations by rearranging Solve quadratic equations by factorising Keywords solve, square root, root Skills check 1 Solve the equation 2x x HELP Section ngel has x CDs in her collection. Write an algebraic expression for the number of CDs that each of these friends has. a my who has twice as many as ngel. b Judith who has four less than my. c ngela who has half as many as Judith. d Jo who has four times as many as ngela. HELP Section 15.1 Solving quadratic equations by rearranging You can solve some quadratic equations by rearranging them to make x the subject. Example 6 Solve the quadratic equation 3x x x x x 5 63 dd 27 to both sides of the equation. Divide both sides by 3. Find the square root of both sides. Remember, when you find the root there are two solutions: positive and negative. Exercise 20F 1 Solve these equations. a r b x c t 2 d y e m f p Find the roots of a 2x b 5y Root is another c 3r 2 5 5r d 3t 2 5 t name for a solution. e y f 2x Quadratic equations

7 Example 7 Solve the equation 2(x 1 3) (x 1 3) (x 1 3) (x 1 3) x x or 7 dd 5 to both sides of the equation. Divide both sides by 2. Take the square root of both sides. Subtract 3 from both sides. Remember to give both solutions. Exercise 20G 1 Find the roots of (r 2 a 2(x 1 1) 2 7)2 5 8 b Solve these equations. a (x 1 1) b t c t d 6 1 3t 2 5 2t e 4(x 1 3) f 7(y 2 2) field is three times as long as it is wide. a Using x for the width of the field, write an expression for its length. b Write an expression for the area Use your answer to part a. of the field, in terms of x. c The field has an area of 1200 m 2. Write an equation for the area of the field. Use your answer to part b. d Solve your equation to find x. e What are the length and the width of the field? 4 Explain why you cannot find a solution to x rectangle has length five times its width. The area of the rectangle is 845 mm 2. What is the width of the rectangle? O2 O3 Solving quadratic equations by factorising Solving quadratic equations by factorising relies on the fact that when a 3 b 5 0, a is 0, b is 0 or both are 0. So if (x 1 2)(x 2 4) 5 0, either x , which If the product of two things is zero, one of them must be zero. means x 5 22, or x , which means x 5 4. To solve a quadratic equation Step 1: Rearrange the equation so that one side is zero. Usually there are two solutions. However, when the expression Step 2: Factorise the quadratic expression. factorises to (x 1 m) 2 5 0, Step 3: Find the solutions. there is only one solution Solving quadratic equations 303

8 Example 8 Solve the equation x 2 5 3x. Step 1: x 2 2 3x 5 0 Step 2: x(x 2 3) 5 0 Step 3: x 5 0 or x So x 5 0 or x 5 3 Subtract 3x from both sides to make one side zero. Factorise the expression. Solve the equation. If the product of two numbers is zero, at least one of the numbers must be zero. Exercise 20H 1 Solve these equations. Factorise first. a x 2 1 7x 5 0 b t 2 2 5t 5 0 c 3x 2 1 6x 5 0 d y 2 5 5y e 0 5 4w w f 5y 5 20y 2 g a 2 a h 5t 5 30t 2 i 14r 5 63r 2 2 Solve these equations. a 2x 2 2 8x 5 0 b 4t 2 1 t 5 0 c 7m m d 8g g e 15f 5 6f 2 f 35w 5 10w 2 Example 9 Find the roots of the equation x 2 2 x Step 1: x 2 2 x Step 2: (x 2 4)(x 1 3) 5 0 Step 3: x or x So x 5 4 or x 5 23 Root is another name for a solution. Subtract 4 from both sides to make one side zero. Factorise the expression. O2 Exercise 20I 1 Find the roots of these equations. a x 2 1 4x b x 2 2 x c x 2 2 6x d x 2 1 x 5 12 e x 2 5 x 1 20 f x 2 1 2x 5 21 g z 2 5 3z 1 4 h 2q 1 q i w 2 5 4w 2 4 j 6t t 2 k 6p p 2 l 10x x 2 2 Jane is three years younger than her older sister. The product of their ages is 54. Use x to represent Jane s age. a Write down an algebraic expression for her sister s age. Remember that Jane is younger. b Write down and simplify an algebraic expression for the product of their ages. c Form and solve an algebraic equation to find the value of x. d Explain why only one of the solutions makes sense. 304 Quadratic equations

9 3 The height of the rectangle is 3 cm more than the width. a Write down an algebraic expression for the height of the rectangle. b Write down an algebraic expression for the area of the rectangle. c Given that the area of the rectangle is 40 cm 2, form and solve a quadratic equation to work out the value of t. t cm 4 rectangular garden is 4 m longer than it is wide. Its area is 165 m 2. a Sketch and label a diagram to show the area. b Form and solve a quadratic equation to work out the dimensions of the garden. 5 I think of a negative number, square it and add five times the original number. My answer is 24. What number did I think of? 6 I think of a positive number. I square it, then subtract six times the number. The answer is 27. What was my original number? O2 O Factorising quadratics of the form ax 2 1 bx 1 c L Why learn this? y breaking down an algebraic expression you can discover some of the properties of the expression. Objectives Solve quadratic equations by factorising * Factorise quadratic expressions of the form ax 2 1 bx 1 c Skills check 1 Write down all the pairs of numbers whose product is a 10 b 12 c Factorise a x 2 1 3x 2 4 b x 2 1 5x 1 6 c x 2 2 8x 1 7 Don t forget negative numbers. Factorising quadratics of the form ax 2 1 bx 1 c In the expression ax 2 1 bx 1 c, the a is the coefficient of x 2 and b is the coefficient of x. In the quadratic expression 3x x 1 4, the coefficient of x 2 is 3. The first terms in the brackets must multiply to give 3x 2. The first terms in the brackets must be 3x and 1x. 3 is a prime number the only factors are 3 and 1. 3x x (3x )(x ) The product of the last two terms must be 14. Possible pairs of numbers are 1 and 4, 21 and 24, 2 and 2, or 22 and 22. The coefficient of x is positive (113), so the two numbers must be positive Factorising quadratics of the form ax 2 1 bx 1 c 305

10 Possible factorisations are (3x 1 4)(x 1 1) (3x 1 1)(x 1 4) (3x 1 2)(x 1 2) Try expanding each one. (3x 1 4)(x 1 1) 5 3x 2 1 3x 1 4x (3x 1 1)(x 1 4) 5 3x x 1 x (3x 1 2)(x 1 2) 5 3x 2 1 6x 1 2x lways check using FOIL to expand the brackets. So 3x x (3x 1 1)(x 1 4) Example 10 Factorise 2x 2 2 7x x 2 2 7x (2x )(x ) Pairs of numbers whose product is 24 are 21 and 4, 1 and 24, or 2 and 22. So the possible factorisations are (2x 2 1)(x 1 4) 5 2x 2 1 8x 2 x 2 4 (2x 1 4)(x 2 1) 5 2x 2 2 2x 1 4x 2 4 (2x 1 1)(x 2 4) 5 2x 2 2 8x 1 x 2 4 (2x 2 4)(x 1 1) 5 2x 2 1 2x 2 4x 2 4 (2x 1 2)(x 2 2) 5 2x 2 2 4x 1 2x 2 4 (2x 2 2)(x 1 2) 5 2x 2 1 4x 2 2x 2 4 Therefore 2x 2 2 7x (2x 1 1)(x 2 4). The only factors of 2 are 1 and 2. One must be positive and one negative since the number term is negative. This gives the 27x required. lways check that the x term is correct. With more practice you will not need to write out all the combinations but will be able to work them out in your head. Exercise 20J Factorise each quadratic expression. 1 2x 2 1 5x x x x x x x x x x 2 2 4x x x x 2 2 5x x x x x x x x 2 2 8x 2 12 Factorising quadratics of the form ax 2 1 bx 1 c when the coefficient of x 2 is not prime When the coefficient of x 2 is not prime, there are more possible cases to consider. 306 Quadratic equations

11 Example 11 Factorise 6x x 1 4. (3x )(2x ) or (6x )(x ) Pairs of numbers whose product is 14 are 2 and 2 or 1 and 4. So the possible factorisations are (3x 1 2)(2x 1 2) 5 1 6x 1 4x x 1 (6x 1 2)(x 1 2) x 1 2x x 1 (3x 1 1)(2x 1 4) x 1 2x x 1 (3x 1 4)(2x 1 1) 5 1 3x 1 8x x 1 (6x 1 1)(x 1 4) 5 (6x 1 4)(x 1 1) 5 This gives 111x required. Therefore 6x x (3x 1 4)(2x 1 1). Factors of 6 are 1 and 6 or 2 and 3. ll terms are positive, so only consider positive numbers. You can stop trying once you have found the correct pair. Exercise 20K 1 Factorise each quadratic expression. a 8x x 1 2 b 4x 2 1 8x 1 3 c 6x x 1 5 d 6x x 1 4 e 8x x 1 12 f 30x x Factorise each quadratic expression. You need to divide through by a common factor first. a 2x 2 1 8x (x ) b 3x x 1 30 c 18x x 1 60 Example 12 Factorise 8x x x x (8x )(x ) or (4x )(2x ) Pairs of numbers whose product is 212 are 212 and 1, 12 and 21, 26 and 2, 6 and 22, 24 and 3, 4 and 23. Possible factorisations are (4x 2 12)(2x 1 1) 5 8x x 2 12 (8x 2 1)(x 1 12) 5 8x x 2 12 (8x 1 6)(x 2 2) 5 8x x 2 12 There are many possible combinations. Try different ones until you find which one will give you 229x. The correct factorisation is (8x 1 3)(x 2 4) 5 8x x O Factorising quadratics of the form ax 2 1 bx 1 c 307

12 Exercise 20L 1 Factorise each quadratic expression. a 10x 2 1 x 2 3 b 12x 2 2 x 2 6 c 4x 2 1 2x 2 6 d 20x x 2 28 e 30x x Factorise each quadratic expression. You need to divide through a 5x 2 1 5x 2 10 b 14x x 2 84 by a common factor first. c 15x x 2 15 d 28x x 1 12 e 24x 2 1 4x 2 4 f 50x x 2 60 g 36x x 1 12 Example 13 Find the values of x which satisfy the equation 8x x 1 4. Step 1: 8x x Rearrange the equation to make one side zero. Step 2: (2x 2 4)(4x 1 1) 5 0 Factorise the equation. Step 3: 2x or 4x Solve the two linear equations. 2x 5 4 or 4x 5 21 So x 5 2 or x O2 O3 Exercise 20M 1 Find the roots of these quadratic equations. Leave your answers as fractions where necessary. a 2a 2 1 5a b 3b 2 1 5b c 4c 2 2 c d 0 5 5d 2 2 8d 2 4 e 6e e f 4f 2 2 6f g 6g g h 0 5 4h 2 1 8h 1 4 i 7i 2 2 3i Find the values of x which satisfy these equations. a 2x 2 5 4x 1 6 b 9x x c 10x x 5 9 d 4x x e 15x x 2 15 f (x 1 2)(x 2 2) 5 3x 3 a Write down an algebraic expression for the area of the rectangle. b The area of the rectangle is 108 cm 2. Form and solve an algebraic equation to find the value of x. c What is the perimeter of the rectangle? 4 I think of a number. Three times the square of my number is equal to twelve times my number. Work out the possible values of my number. 3x 2x Quadratic equations

13 5 Next year Yvette will be four times her daughter melia s age. Let x represent melia s age next year. a Write down an algebraic expression for i Yvette s age next year ii melia s age this year iii Yvette s age this year. b The product of their ages is 351. Form and solve an algebraic equation to work out melia s age. c How old is Yvette this year? * O L Why learn this? This method solves quadratic equations that you can t factorise, like x 2 1 3x 2 7. Using the quadratic formula Objectives * Solve quadratic equations by using the quadratic formula * Decide how many solutions a quadratic equation has by considering the discriminant Skills check 1 Using the formula x 5 3y 8 2 z, find the value of x when a y 5 16, z b y 5 24, z 5 49 c y 5 80, z The quadratic formula Sometimes a quadratic expression cannot be factorised. You can use the quadratic formula to solve a quadratic equation of the form ax 2 1 bx 1 c 5 0, where a 0. x 5 2b 6 b 2 2 4ac 2a Keywords quadratic formula, discriminant There will be two solutions. You do not need to learn the formula it will be on the exam formula sheet. e careful! You cannot use the quadratic formula until you have made one side of the equation zero. Example 14 Use the quadratic formula to solve the equation x 2 1 3x a 5 1, b 5 3, c 5 27 x 5 2b 6 b 2 2 4ac 2a x or x Write down the values of a, b and c. Simplify the calculation. Follow the order of operations. Substitute these values into the quadratic formula. e careful with the negative value. Leave your answer in surd form Using the quadratic formula 309

14 Exercise 20N Use the quadratic formula to solve each equation. Leave your answers in surd form. 1 x 2 1 3x x 2 1 5x x 2 1 6x x 2 1 6x x 2 2 2x x 2 1 8x y y r 2 2 8r t 2 2 2t g 2 1 7g 1 3g 5 0 Example 15 Solve the quadratic equation 2x 2 5 6x x 2 2 6x a 5 2, b 5 26, c x 5 2b 6 b 2 2 4ac 2a 5 2 (26) 6 (26) (212) First rearrange the equation to make one side zero. e very careful with positive and negative numbers Divide all terms by 2. O2 * O3 Exercise 20O Make sure you rearrange 1 Use the quadratic formula to solve these equations. the equations first. Leave your answers in surd form. a x 2 5 4x 1 1 b x x c x 2 2 8x 5 6 d x x e 4 1 2x 5 x 2 f x 2 1 8x Try to solve this quadratic equation using the quadratic formula. x 2 1 x Explain why you cannot find a solution. 3 carpet manufacturer wishes to make carpet tiles with area 1500 cm 2. The tiles are rectangular and the length is 10 cm less than double the width. Work out the dimensions of a carpet tile. 4 Look at your answer to Q3. Did you need to use the quadratic formula? Explain your answer. 310 Quadratic equations

15 The discriminant Question 2 in Exercise 20O asked you to try to solve the equation x 2 1 x Using the quadratic formula a 5 1, b 5 1, c 5 1 x 5 2b 6 b 2 2 4ac 2a The calculations result in trying to find the square root of a negative number. This has no real solutions you will learn more about this if you do -level maths. b 2 2 4ac in the quadratic formula is known as the discriminant. In general, when b 2 2 4ac. 0, there are two distinct solutions to the quadratic equation when b 2 2 4ac, 0, there are no real solutions to the quadratic equation when b 2 2 4ac 5 0, there is one solution (sometimes called a repeated root). Example 16 y considering the discriminant, decide whether each of these quadratic equations has zero, one or two solutions. a 3x 2 1 2x b 7x x 2 8 c 9x x * a 3x 2 1 2x a 5 3, b 5 2, c 5 25 Write down the values of a, b and c. b 2 2 4ac Work out b 2 2 4ac Since there are two solutions. b 7x x 2 8 7x x Rearrange to the form ax 2 1 bx 1 c 5 0 a 5 7, b 5 210, c 5 8 b 2 2 4ac 5 ( 10) Since 2124, 0 there are no solutions Using the quadratic formula 311

16 c 9x x 9x x Rearrange to the form ax 2 1 bx 1 c 5 0 a 5 9, b 5 224, c 5 16 b 2 2 4ac 5 (224) There is one (repeated) solution. * Exercise 20P 1 For each quadratic equation, decide if there are zero, one or two solutions. a 3x 2 1 2x b 5m 2 1 9m c 3t 2 1 6t d 4d 2 2 5d e 0 5 2z 2 1 5z 1 1 f 4x 2 5 3x 2 1 g 9t 5 5t h 2q q L Why learn this? In mathematics, as in life, it is important to have more than one way to solve a problem. Completing the square Objectives * Solve a quadratic equation by completing the square Skills check HELP Section Expand and simplify a (x 1 2) 2 b (x 2 3) 2 c (x 2 5) 2 Completing the square Completing the square is another way to solve a quadratic equation which cannot be factorised. Expanding an expression of the form (x 1 a) 2 gives (x 1 a)(x 1 a) 5 x 2 1 2ax 1 a 2 Working backwards, this can be used to complete the square. Consider the equation x 2 1 4x For the coefficient of x to be 4 the squared bracket must be (x 1 2) 2. ut (x 1 2) 2 5 x 2 1 4x 1 4. This is half the coefficient of x. To get from (x 1 2) 2 to x 2 1 4x 1 10 you need to subtract 4 and then add 10. x 2 1 4x (x 1 2) (x 1 2) Quadratic equations

17 Example 17 Write the expression x x 1 9 in completed square form. (x 1 5) 2 5 x x 1 25 The expression required is x x (x x 1 25) (x 1 5) (x 1 5) Halve the coefficient of x. Subtract the square of the number in the bracket. Put in the original number term. * Exercise 20Q 1 Write each expression in completed square form. a x 2 1 6x 1 3 b x 2 1 2x 1 7 c x 2 2 8x 1 5 d x x 1 12 e x 2 2 4x 2 7 f x x 2 1 * 2 Write each algebraic expressions in the form e careful with the values of p and q. (x 1 p) 2 1 q, giving the values of p and q. re they positive or negative? a x x 1 32 b x 2 1 2x 1 2 c x 2 2 4x 1 20 d x x 1 10 e x 2 2 6x 2 3 f x 2 2 4x 2 2 Example 18 y completing the square, solve the equation x 2 2 8x Leave your answer in surd form. * x 2 2 8x (x 2 4) (x 2 4) So (x 2 4) (x 2 4) x x Write the expression in completed square form. Solve the equation by rearranging. This is the exact answer in surd form. Exercise 20R 1 Solve the quadratic equations by completing the square. a x x b x 2 1 2x c x 2 2 8x d x x e x 2 2 4x f x 2 1 6x Completing the square * 313

18 * 2 Give the exact solution to these quadratic equations by completing the square. a x 2 1 8x b x 2 1 4x c x 2 2 2x d x 2 2 8x e x x f x x Leave your answers in surd form where apprpriate. 3 Solve these quadratic equations Don t forget to rearrange the equations first. by completing the square. a x 2 5 6x 2 4 b 3x(x 1 6)5 6 c (x 1 1)(x 2 5) 5 7 d (x 2 2)(x 1 8) 5 7 e 2 x x f 3 (r 2 1)(r 1 2) 5 1 Review exercise 1 Factorise the expression x 2 1 6x 1 5. [2 marks] 2 I think of a number, square it, then subtract three times the number. The result is 108. Form and solve an algebraic equation to work out the possible values of the number I thought of. [4 marks] O3 3 I think of a number, square it, then add it to 5 times the number. The answer is 24. Form and solve an algebraic equation to work out the possible values of the number I thought of. [4 marks] 4 a Factorise 2x x 2 8. [2 marks] b Hence solve the equation 2x x [2 marks] 5 Factorise 6y y 2 5. [2 marks] 6 Use the quadratic formula to solve 2x 2 2 6x Leave your answer in surd form. [3 marks] 7 a Factorise the quadratic expression 6x x [2 marks] b Hence solve the equation 6x x Leave your answers as fractions. [2 marks] * 8 rectangular piece of land has length 3 m more than double the width. The area of the rectangle is 170 m 2. Work out the dimensions of the rectangle. [5 marks] O3 9 rectangular rug is 6 m longer than its width. The area of the rug is 16 m 2. Calculate the dimensions of the rug. [5 marks] 314 Quadratic equations

19 10 a Find the values of m and n such that x 2 1 4x (x 1 m) 2 2 n. [2 marks] b Hence solve the equation x 2 1 4x by rearranging, leaving your answer in the form a 6 b. [3 marks] * 11 How many roots does each of these quadratic equations have? a 5x 2 2 2x b 3x x c 4x x [6 marks] 12 a Write the following algebraic expression in completed square form. x 2 2 4x 1 2 b Hence find the exact solution to the equation x 2 2 4x [2 marks] [2 marks] Chapter summary In this chapter you have learned how to factorise a quadratic expression of the form x 2 1 bx 1 c solve quadratic equations by rearranging factorise a quadratic expression that is the difference of two squares solve quadratic equations by factorising factorise quadratic expressions of the form ax 2 1 bx 1 c * solve quadratic equations by using the quadratic formula * decide how many solutions a quadratic equation has by considering the discriminant * solve a quadratic equation by completing the square * Chapter 20 Summary 315