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Number and Algebra Linear and non-linear relationships Solve linear equations involving simple algebraic fractions. • solve a wide range of linear equations, including those involving one or two simple algebraic fractions, and check solutions by substitution. • represent word problems, including those involving fractions, as equations and solve them to answer the question. Solve simple quadratic equations using a range of strategies. • use a variety of techniques to solve quadratic equations, including grouping, completing the square, the quadratic formula and choosing two integers with the required product and sum. Quadratic - the highest power of the variable is two. i u2 + 2P = 2h Air pressure and the Bernouilli effect. s = ut + ½at2 Distance travelled by accelerating object. ∂u + ∇2 u + v( x ) u = 0 ∂t Design of integrated circuits in mobile phones and computers. A TASK A graphics calculator can be used to solve quadratic equations. Learn how to use a graphics calculator to solve quadratic equations such as: 6x2 + 7x − 5 = 0 The solutions are x=0.5, and x = ˉ1.67 A LITTLE BIT OF HISTORY The Babylonians (about 400 BC) appear to be the first to solve quadratic equations. The Hindu mathematician Brahmagupta (598-665 AD) essentially used the quadratic formula to solve quadratics. Brahmagupta used letters for variables and considered negative numbers. al-Khwarizmi (c 820 AD) also essentially used the quadratic formula to solve quadratics but didn't consider negative solutions. ax 2 + bx + c = 0 x= −b ± b 2 − 4ac 2a Abraham bar Hiyya Ha-Nasi was the first to publish a book in 1145 which used the complete quadratic formula. 87 Solving Linear Equations To solve an equation is to find the value of the unknown number (the variable) in the equation. The formal way to solve an equation is to choose to do the same thing to each side of the equation until only the unknown number remains on one side of the equation. Exercise 7.1 Solve each of the following equations: x + 7 =5 x + 7 − 7= 5 − 7 x = ˉ2 c – 4 =7 c – 4 + 4 = 7 + 4 c = 11 1 4 7 x + 8 = 3 t ÷ 4 = 3 z ÷ 7 = 11 d ÷ 3 =5 d ÷ 3 × 3= 5 × 3 d = 15 2 b – 7 = ˉ11 5 a + 11 = 43 8 ˉ2w = 48 3 6 9 + − × ÷ is is is is − + ÷ × 5y = 25 21m = 105 x – 29 = 41 11 13 15 17 19 21 23 ˉ6(m − 2) = 18 4(x + 2) = 16 ˉ2(f + 5) = 10 4(p − 3) = ˉ20 5(a + 6) = 25 7(x − 3) = ˉ28 ˉ3(x + 5) = 12 ˉ5(2x − 3)= 25 ˉ10x + 15= 25 ˉ10x = 25 − 15 ˉ10x= 10 x = 10 ÷ ˉ10 x = ˉ1 24 26 28 30 32 34 Check: ˉ5(2×ˉ1 − 3) = ˉ5×ˉ5 = 25 36 3(x − 4) = 12 ˉ2(x − 3) = 14 3(x + 5) = ˉ18 4(2x − 2) = 16 ˉ5(2d + 7) = 65 5(2x − 3) = ˉ15 7(2x − 1) = 21 25 27 29 31 33 35 37 2(2x − 5) = 10 6(2x + 4) = ˉ36 ˉ4(5x − 4) = 24 2(2x − 5) = ˉ10 7(ˉ3x + 2) = 35 3(5x − 4) = ˉ15 5(ˉ5x − 4) = 20 38 39 40 41 42 43 44 45 Check: 3(ˉ2×0.8 − 3) − 4×0.8 3×ˉ4.6 − 3.2 = ˉ17 46 2(x – 3) + 3x = 9 If the unknown occurs 5(x – 3) – 2x = 9 more than once, put 3(x + 2) + 2x = 11 the unknowns together 3(2x – 2) + 4x = 14 (5x−3x=2x). 3x + 2(x + 2) = ˉ11 x + ˉ3(x – 1) = 5 2x + 3(x – 1) + 3x = 13 5x +2(2x + 3) – x + 5 = 35 ˉ9x + 5x + 3(1 – 2x) + 5 = 29 Check: 4(ˉ2 − 3) = 4×ˉ5 = ˉ20 3(ˉ2x − 3) − 4x = ˉ17 ˉ6x − 9 − 4x = ˉ17 ˉ10x − 9= ˉ17 ˉ10x = ˉ17 + 9 ˉ10x = ˉ8 x = ˉ8 ÷ ˉ10 x = 0.8 10 12 14 16 18 20 22 The inverse of The inverse of The inverse of The inverse of 3(x + 4) = 21 2(a − 3) = 4 2(x + 3) = ˉ16 3(t − 5) = 12 5(b + 4) = 15 ˉ4(n − 7) = 12 4(n − 7) = 12 4(d − 3) = ˉ20 4d − 12 = ˉ20 4d = ˉ20 + 12 4d = ˉ8 d = ˉ8 ÷ 4 d = ˉ2 88 5a = ˉ15 5a ÷ 5 = ˉ15 ÷ 5 a = ˉ3 Solving Linear Equations 5x − 3 = 2x + 9 5x − 3 − 2x = 9 3x − 3 = 9 3x =9+3 3x = 12 x = 12 ÷ 3 x = 4 Check: (0.36)2 + 8×0.36 − 3 = 0 x + 2 = −5 3 Multiply by the denominator of the fraction. x a 57 7 + 2 = 3 x Check: ˉ21/3 + 2 = ˉ7 + 2 = ˉ5 2x − 3 − = 2 5 59 6 = 4 −1 x −1 =1 2 x−2 − = 1 63 2 x+6 − = 3 65 2 61 2x − 3= ˉ2×5 2x − 3= ˉ10 2x = ˉ10 + 3 2x = ˉ7 x = ˉ7÷2 x = ˉ3.5 Check: (2×ˉ3.5−3)/5 = ˉ10/5 = ˉ2 67 69 x 2x − − = 2 2 3 {×6} Check: 12/2−2×12/3=6−8 = ˉ2 2x + 1 x = −4 3 2 2(2x+1)= 3x − 24 4x + 2= 3x − 24 4x − 3x= −24 − 2 x = ˉ26 If the unknown is on 5x − 8 = 2x + 4 both sides of the =, 2x + 2 = 9 − 5x put the unknowns 5h + 4 = 4h + 7 together . 6y + 3 = y + 28 5w − 4 = 5 − 4w 4(t − 2) = 2t + 4 3(m + 3) + 2 = 15 − m 4x + 5 − x = 2(3 − x) + 4 − 55 4 + 3 = 2 x + 6= ˉ15 x = ˉ15 − 6 x = ˉ21 3x − 4x= ˉ12 ˉx= ˉ12 x = 12 47 48 49 50 51 52 53 54 71 73 75 77 {×6} Check: (2×ˉ26+1)/3 = ˉ26/2−4 (ˉ52+1)/3 = ˉ13−4 ˉ17 = ˉ17 79 81 83 85 2x +5= 2 3 x −5 =1 2 3x − 2 =3 4 x 3 + =1 2 5 2x x − =3 3 2 x x = +4 2 3 x 2x +3= 2 3 2x + 1 x + =5 3 2 3x − 4 x = −1 2 5 2x 5 − + = 3x 3 2 a 56 5 − 2 = 4 y − 58 3 − 3 = 1 x − 60 5 −15 = 3 x+2 =1 3 x −5 − = 2 64 3 x+6 =2 66 3 62 3a 68 4 − 2 = 3 70 72 74 76 78 80 82 84 86 b−2 − = 2 3 2b + 4 =2 3 x 1 − =2 3 4 3x x − =2 2 3 x x −1 = 4 3 − x 3x +1 = 2 3 3x − 2 x = +6 2 3 3x − 1 x = +4 5 2 3x x x − − =2 2 3 4 Chapter 7 Solving Equations 89 Solving Linear Equations Each day, in Australia, millions of real-world problems are solved by the use of linear equations. The basic idea is to: 1 Write the formula. 2 Substitute. 3 Solve for the unknown. Exercise 7.2 The circumference, C, of a circle is given by the formula: C = 2πr, where r is the radius of the circle. A circular horse yard is to have a circumference of 56 m. What should be the radius of the horse yard? Expressions such as C = 2πr C = 2πr {write the formula} are linear because the highest power of r is 1 (r = r1). 56 = 2πr {substitute C = 56} 56÷(2π) = r {inverse of × is ÷} 8.91= r If you don't use the '(' and ')' Radius of horse yard = 8.91 m. on your calculator you will Check: 2πr = 2π×8.91 = 55.98 m 1 90 get 56÷(2π) wrong. The circumference, C, of a circle is given by the formula: C = 2πr, where r is the radius of the circle. A circular horse yard is to have a circumference of 75 m. What should be the radius of the horse yard? 2 The perimeter, P, of a rectangle is given by P = 2(l + b), where l is the length and b is the breadth. If the length of a house block is 43 m and the perimeter is 138 m, what is the breadth of the house block? 3 Speed, v, is given by the formula: v = s ÷ t ( v = t ), where s is the distance and t is the time. If thunder is heard eight seconds after the lightning is seen, how far away was the lightning (Assume sound travels at 330 m/s)? 4 The sum of the interior angles of a polygon is given by the formula: S = 90(2n − 4), where n is the number of sides on the polygon. How many sides in a polygon with an interior angle sum of 540°? 5 The volume, V, of a square based prism is given by the formula: V = w2h, where w is the width of the base and h is the height of the prism. If the width of the prism is 4 cm and the volume is 96 cm3, what is the height of the prism? 6 The volume of a cylinder, V, is given by the formula: V = πr2h, where r is the radius of the base of the cylinder and h is the height of the cylinder. If a cylinder with a base radius of 6.5 cm has a volume of 1487 cm3, what is the height of the cylinder? 7 The volume of a cone, a circular based pyramid, is given by the formula: V= cone. If a cone has a radius of 1.1 m and a volume of 6.8 m3, what is the height of the cone? s πr 2 h 3 where r is the radius of the base of the cone and h is the height of the Exercise 7.3 Five times a number increased by four is divided by three to obtain twentytwo. What is the number? 1 2 3 4 Three times a number increased by five is divided by two to obtain thirteen. What is the number? Four times a number decreased by six is divided by seven to obtain twenty-two. What is the number? Three consecutive numbers have a sum of forty-five. What are the numbers (4,5, and 6 are consecutive numbers)? Find three consecutive numbers whose sum is six times the first number. A rectangle has a length which is 5 m less than twice its width. What is the width of the rectangle, if the perimeter is 80 m? 5 A rectangle has a length which is 3 m less than twice its width. What is the width of the rectangle, if the perimeter is 42 m? Let the width be w, 2×w +2×(w − 5)= 80 2w + 2w − 10= 80 4w − 10= 80 4w = 80 + 10 4w = 90 w = 90 ÷ 4 w = 22.5 m (the width) 6 A rectangle has a length twice the size of the width. What is the width of the rectangle if the perimeter is 180 cm? 7 An isosceles triangle has two angles each four times the size of the third angle. What is the size of each angle? 8 The crocodile was described as being one-third tail, one-quarter head and with a 200 cm body. What was the length of the crocodile? 9 The first side of a triangle is 20 cm longer that the second side. The third side is half the length of the second side. What is the length of the second side if the perimeter is 300 cm? Let the number be x, 5x + 4 = 22 3 5x + 4 5x + 4 5x 5x x x = 22×3 = 66 = 66 − 4 = 62 = 62 ÷ 5 = 12.4 Check: (5×12.4 + 4)/3 = 66/3 = 22 Check: 2×22.5+2(22.5−5) = 80 The crocodile was described as being one-third tail, one-quarter head and with a 300 cm body. What was the length of the crocodile? Let the length of the crocodile be x, 1 1 x + x + 300 = x 3 4 4x + 3x + 3600 = 12x {×12} 7x + 3600 = 12x 3600 = 12x − 7x 3600 = 5x 10 The difference between two 3600 ÷ 5 = x numbers is 29. One number is 5 x = 720 (crocodile was 7.2 m) less than three times the other. Check: 720/3+720/4+300 = 720 What are the two numbers? Chapter 7 Solving Equations 91 Solving Quadratics This is a quadratic because the highest power of the unknown, x,is two. ax2 + bx + c = 0 Thousands of real life problems can be written as quadratics. x2 suggests that there could be none, one, or two solutions. Exercise 7.4 Solve the following quadratics: x2 + 6x + 9 = 0 (x + 3)2 = 0 {perfect square} x + 3 = 0 {square root both sides} x = ˉ3 {inverse of +3} Check: (ˉ3)2 + 6×ˉ3 + 9 = 0 x2 − 8x + 16 = 0 (x − 4)2= 0 x − 4 = 0 x = 4 {perfect square} {square root both sides} {inverse of −4} Check: (4)2 − 8×4 + 16 = 0 1x2 + 2x + 1 = 0 2x2 + 4x + 4 = 0 3x2 + 8x + 16 = 0 4x2 + 10x + 25 = 0 5x2 + 14x + 49 = 0 6x2 − 2x + 1 = 0 7x2 − 6x + 9 = 0 8x2 − 8x + 16 = 0 9x2 − 10x + 25 = 0 10x2 − 18x + 81 = 0 4x2 + 12x + 9 =0 2 (2x) + 2×2x×3 + (3)2 = 0 (2x + 3)2 = 0 2x + 3 = 0 2x = ˉ3 x = ˉ1.5 114x2 + 12x + 9 = 0 12 9a2 + 6x + 1 = 0 134x2 + 8x + 4 = 0 144x2 − 4x + 1 = 0 15 9y2 + 24y + 16 = 0 16 9x2 − 12x + 4 = 0 17 25x2 − 30x + 9 = 0 x2 + 6x + 8 = 0 2 x + 4x + 2x + 8 = 0 {grouping pairs} x(x + 4) + 2(x + 4)= 0 {factorising } (x + 4)(x + 2) = 0 {factorising} Either x + 4 = 0 or x + 2 = 0 x = ˉ4 or x = ˉ2 18x2 + 5x + 4 = 0 19x2 + 3x + 2 = 0 20x2 + 5x + 6 = 0 21x2 + 7x + 6 = 0 22x2 + 8x + 12 = 0 23x2 + 8x + 15 = 0 24x2 + 6x + 8 = 0 25x2 + 9x + 14 = 0 {perfect square} {square root both sides} {inverse of +3} {inverse of ×2} 2 Check: 4×(ˉ1.5) + 12×ˉ1.5 + 9 = 0 Check: (ˉ4)2 + 6×ˉ4 + 8 = 0 Check: (ˉ2)2 + 6×ˉ2 + 8 = 0 x2 − 3x − 10 = 0 x2 − 5x + 2x − 10 = 0 {grouping pairs} x(x − 5) + 2(x − 5)= 0 {factorising } (x − 5)(x + 2) = 0 {factorising} Either x − 5 = 0 or x + 2 = 0 x = 5 or x = ˉ2 Check: (5)2 − 4×5 − 10 = 0 Check: (ˉ2)2 − 4×ˉ2 − 10 = 0 92 Solving means find the values of x that makes this true. 26x2 + x − 6 = 0 27x2 + 5x − 6 = 0 28x2 − 2x − 8 = 0 29x2 − 5x − 6 = 0 30x2 − 4x − 12 = 0 31x2 − 6x + 9 = 0 32x2 − 10x + 9 = 0 33x2 − 8x + 12 = 0 Use a2 + 2ab + b2 = (a + b)2 to solve quadratics. Completing the Square x2 + 6x + 2 = 0 = ˉ2 x2 + 6x 2 x + 6x + 9 = ˉ2 + 9 (x + 3)2 = 7 x + 3 = ±√7 x =ˉ3±√7 x = ˉ0.35 or x = ˉ5.65 x2 + 6x + 9 = (x + 3)2 Add 9 to both sides to complete the square. (√7)2 = 7 and (−√7)2 = 7 Which is why there are two solutions: +√7 or −√7 thus ±√7 Exercise 7.5 Solve the following quadratics by completing the square: x2 + 8x − 3 = 0 = 3 {move constant term} x2 + 8x x2 + 8x + 16 = (x + 4)2 x2 + 8x + 16 = 3 + 16 (x + 4)2 = 19 x + 4 = ±√19 x =ˉ4±√19 x = 0.36 or x = ˉ8.36 {+16 to complete square} {perfect square} {square root both sides} {inverse of +4} {use calculator} Check: (0.36)2 + 8×0.36 − 3 = 0 Check: (ˉ8.36)2 + 8×ˉ8.36 − 3 = 0 x2 − 10x + 1 = 0 x2 − 10x = ˉ1 {move constant term} x2 − 10x + 25 = (x − 5)2 x2 − 10x + 25= ˉ1 + 25 (x − 5)2 = 24 x − 5 = ±√24 x = 5±√24 x = 9.90 or x = 0.10 {+25 to complete square} {perfect square} {square root both sides} {inverse of +4} {use calculator} Check: (9.90)2 − 10×9.90 + 1 = 0 Check: (0.10)2 − 10×0.10 + 1 = 0 x2 − 3x − 4 = 0 = 4 x2 − 3x x2 − 3x + (3/2)2 = (x − 3/2)2 x2 − 3x + (1.5)2= 4 + (1.5)2 (x − 1.5)2 = 6.25 x − 1.5 = ±2.5 x = 1.5±2.5 x = 4 or x = ˉ1 Check: (9.90)2 − 10×9.90 + 1 = 0 Check: (0.10)2 − 10×0.10 + 1 = 0 {move constant term} {3/2 = half middle number} {complete square, 3/2=1.5} {perfect square} {square root both sides} {inverse of +4} 1x2 + 6x − 1 = 0 2x2 + 2x − 3 = 0 3x2 + 4x − 4 = 0 4x2 + 4x − 6 = 0 5x2 + 6x − 2 = 0 6x2 + 10x − 5 = 0 7x2 + 2x + 1 = 0 8x2 + 6x + 4 = 0 9x2 + 8x + 3 = 0 10x2 + 12x + 2 = 0 11x2 − 2x − 3 = 0 12x2 − 4x − 1 = 0 13x2 − 6x − 5 = 0 14x2 − 8x − 7 = 0 15x2 − 10x − 2 = 0 16x2 − 12x − 4 = 0 17x2 − 14x + 4 = 0 18x2 − 6x + 2 = 0 19x2 − 4x + 1 = 0 20x2 − 8x + 3 = 0 21x2 − 3x − 1 = 0 22x2 + 3x − 3 = 0 23x2 + 5x − 5 = 0 24x2 − 5x − 2 = 0 25x2 + 7x − 4 = 0 26x2 − 7x − 3 = 0 27x2 + 9x + 1 = 0 28x2 − 9x + 2 = 0 29x2 + 11x + 5 = 0 30x2 − 11x + 2 = 0 Chapter 7 Solving Equations 93 The Quadratic Formula Thousands of problems can be expressed as quadratics. Quadratics can be solved by guess and check, sketching graphs, factorising, and using the quadratic formula. Given the quadratic: The solution is: ax2 + bx + c = 0 This is the world famous quadratic formula. −b ± b 2 − 4ac x= 2a Exercise 7.6 Use the quadratic formula to solve the following quadratics: x2 + x − 2 = 0 x2 + x − 2 = 0 1 Write the quadratic ax2 + bx + c = 0 2 Write the general quadratic a=1 b=1 c=ˉ2 3 Decide values of a, b, c 4 Write the quadratic formula x= 5 Substitute for a, b, and c x= 6 Calculate x= x= x= x= Check: x=1, 12 + 1 − 2 = 0 Check: x=ˉ2, (ˉ2)2 + ˉ2 − 2 = 0 = 4 − 4 = 0 1x2 – x – 2 = 0 2x2 + x – 2 = 0 3x2 – 2x – 3 = 0 4x2 + 2x + 1 = 0 5x2 + 3x + 2 = 0 6x2 + 5x + 6 = 0 7x2 + 4x + 3 = 0 8 x2 – 2x + 1 = 0 9x2 – 3x + 2 = 0 10x2 – 5x + 6 = 0 94 −b ± b 2 − 4ac 2a 1 ± 12 − 4 ×1× − 2 2 ×1 − − 1± 1+ 8 2 − 1± 9 2 − 1± 3 2 ± means + or − − − 1− 3 1+ 3 or x = 2 2 x = 1 or x = ˉ2 which is the same as which is the same as which is the same as which is the same as which is the same as which is the same as which is the same as which is the same as which is the same as which is the same as The numerator needs brackets if using a calculator. (x – 2)(x + 1) = 0 (x – 1)(x + 2) = 0 (x + 1)(x – 3) = 0 (x + 1)(x + 1) = 0 (x + 2)(x + 1) = 0 (x + 2)(x + 3) = 0 (x + 1)(x + 3) = 0 (x – 1)(x – 1) = 0 (x – 1)(x – 2) = 0 (x – 2)(x – 3) = 0 The Quadratic Formula y = 2x2 + 5x − 7 is called a quadratic equation because the highest power of x is 2. Exercise 7.7 Use the quadratic formula to solve the following quadratics: 5x2 − 2x − 4 = 0 5x2 − 2x − 4 = 0 1 Write the quadratic ax2 + bx + c = 0 2 Write the general quadratic a=5 b=ˉ2 c=ˉ4 3 Decide values of a, b, c 4 Write the quadratic formula x= −b ± b 2 − 4ac 2a 5 Substitute for a, b, and c x= −− 2 ± ( − 2) 2 − 4 × 5 × − 4 2×5 6 Calculate x= 2 ± 9.17 10 x= (2 + 9.17) 10 Check: x=1.12, 5×1.122 − 2×1.12 − 4 = 0 Check: x=ˉ0.72, 5×(ˉ0.72)2 − 2×ˉ0.72 − 4 = 0 See Technology 7.1 or x = (2 + 9.17) 10 x = 1.12 or x = ˉ0.72 1 2x2 + 3x – 2 = 0 2x2 + 2x – 5 = 0 44x2 – 3x – 3 = 0 3 5x2 – 2x – 1 = 0 6x2 – 3x – 5 = 0 5 2x2 + x – 2 = 0 87x2 – 2x – 5 = 0 7x2 + 5x + 1 = 0 10x2 – 11x – 1 = 0 94x2 + 8x + 3 = 0 124x2 – 5x = 0 11 6x2 + 8x – 4 = 0 Nothing is all wrong. 144x2 – 8 = 0 13x2 + 3x + 2 = 0 Even a busted clock is 16 5x2 – 4x = 0 158x2 + 3x= 0 right twice a day. 1814x2 – 5 = 0 17 16x2 – 4= 0 20 ˉ4x2 – 3x + 1 = 0 19 ˉx2 + 5x – 2 = 0 22 ˉ3x2 + 3x + 1 = 0 21 ˉ2x2 + 3x + 5 = 0 24 ˉ3x2 – 6x – 2 = 0 23 ˉx2 – 4x – 1 = 0 25 A stone is thrown vertically with a velocity of 30 m/s. The motion of the stone is described by the relationship: 20 m 5t2 - 30t + h = 0, where h is the height in metres and t is the time in seconds. How long will it take the stone to reach a height of 20 m? 26 Calculate x in each of the following figures: Perimeter=60 a) b) c) 5 x+2 x x+2 Area=5 x x+15 3x x2 Chapter 7 Solving Equations 95 Mental Computation Mental computation gives you practice in thinking. Exercise 7.8 1 Spell Quadratic. 2 What is the quadratic formula? 3 3x2 + 2x − 2 = 0. What are the values of a, b, and c? 4 Expand (x − 2)2 5 Factorise x2 + 4x + 3 6 What is the value of: Log10100 7 Solve: x + y = 10, xy = 21 8 ˉ2 − ˉ5 6 + 6×30/100 9(x−3)4 = 6 + 180/100 10 Increase $6 by 30% = 6 + 1.8 x2 + 4x + 3 3+1=4 3×1=3 = (x+3)(x+1) = $7.80 Exercise 7.9 1 Spell 'Solving linear equations'. 2 What is the quadratic formula? 3 5x2 − x − 3 = 0. What are the values of a, b, and c? 4 Expand (x + 2)2 5 Factorise x2 + 7x + 10 Learning without thought is labor lost; thought without 6 What is the value of: Log101000 learning is perilous - Confucius. 7 Solve: x + y = 11, xy = 30 8 ˉ2 − 3 9(x−2)4 10 Increase $7 by 30% War does not determine who is right, only who is left - a paraprosdokian. Exercise 7.10 1 Spell 'Completing the square'. 2 What is the quadratic formula? 3x2 − 2x + 4 = 0. What are the values of a, b, and c? 4 Expand (x − 3)2 5 Factorise x2 + 5x + 4 6 What is the value of: Log1010 000 7 Solve: x + y = 6, xy = 8 You do not need a parachute to skydive. 8 ˉ7 + ˉ7 You only need a parachute to skydive twice - another paraprosdokian. 9(x−2)5 10 Increase $8 by 30% Mechanical engineers design and supervise the construction and operation of machinery. • Relevant school subjects are English, Mathematics, and Physics. • Courses usually involve an engineering degree. 96 Build maths muscle and prepare for mathematics competitions at the same time. Competition Questions Exercise 7.11 If 48a = b2 and a and b are positive integers, find the smallest value of a. 48a = b2 16×3a = b2 42×3a = b2 y= 3x − 5 2 thus a = 3 If a and b are positive integers, find the smallest value of a: 1 28a = b2 2 24a = b2 3 63a = b2 What is the value of x when y = ˉ2? What is the value of x when y = ˉ3? ˉ3×2 = 3x − 5 ˉ6 + 5= 3x ˉ1 ÷ 3= x x = ˉ⅓ or ˉ0.33 4 y = 4x + 3 5 y= 2x 1 + 5 3 6 y= 4x + 3 5 (2x − 3)(x + 5) = 0 Either 2x − 3= 0 2x = 3 x = 1.5 Check: (2×ˉ5−3)(ˉ5+5) Check: (2×1.5−3)(1.5+5) = ˉ13×0 = 0×6.5 =0 =0 Or Solve each of the following quadratic equations: 7 (x + 1)(x −1) = 0 x + 5 = 0 x = ˉ5 8 (x − 2)(2x − 1) = 0 9 (3x + 1)(x − 2) = 0 10 (2x − 2)(x + 4) = 0 11 What is the equation of each of the following quadratics: x-intercepts x = ˉ1 thus x + 1 = 0 x = 2 thus x − 2 = 0 5 4 3 2 1 0 -2 -1 -1 0 1 2 3 y = (x + 1)(x − 2) or y = x2 − x − 2 -2 -3 a) b) 5 6 5 4 4 3 3 2 2 1 1 -3 0 -4 -3 -2 -1 -1 -2 -3 0 1 2 3 -2 -1 0 -1 0 1 2 3 4 5 -2 -3 -4 -5 Chapter 7 Solving Equations 97 A Couple of Puzzles Exercise 7.12 1 On Monday a plant is 2 cm high. Each day the plant doubles its height from the day before. How high will the plant be on Friday? 2 Can you cut this shape into a) Two conguent shapes? b) Three conguent shapes? c) Four congruent shapes? d) Six congruent shapes? A Game Calculator Hi Lo is played on a calculator. You try to guess the secret number in as few guesses as possible. 1 Someone else gives the limits of the secret number eg., 1 to 10, 1 to 100, 1 to 1000 and enters the secret number in the calculator's memory and clears the display. 2 You enter a guess on the calculator. 3 4 Someone else divides your number by the number in memory. If the display is: greater than 1 then the guess was too high. less than 1 then the guess was too low. equal to 1 then the guess was correct. ÷ 74 STO 50 RCL M+ = 13.867321 0.67567567 Try to guess the secret number in as few guesses as possible. A Sweet Trick Impress your audience by showing that you know some tricky patterns. 98 37×3 37×6 37×9 37×12 37×15 = = = = = 1×8 + 1 = 12×8 + 2 = 123×8 + 3 = 1234×8 + 4 = 12345×8 + 5 = Think about your presentation: • Think up some story about these numbers? • Audience has calculators, you don't have a calculator? 92 − 22= 892 − 122= 8892 − 1122= 88892 − 11122= 888892 − 111122= 62 − 52= 562 − 452= 5562 − 4452= 55562 − 44452= 555562 − 444452= Investigations Investigation 7.1 A quadratic function The distance an object falls in a certain time, when dropped, can be modelled by the quadratic function: s = 4.9t2, where s is the distance in metres an object falls in t seconds. Investigate Example: A stone takes 2.75 seconds to reach the bottom of a well. s = 4.9t2 Depth of well = 4.9x2.752 = 37 m Ways of testing the function A GPS unit and a stone jump off a cliff at the same time. Which will hit the ground first? s = 4.9t2 The GPS because it doesn't have to ask for directions. Investigation 7.2 A linear function? 1 Turn on a tap and let the water run at a constant rate. 2 Ready the bucket by putting a ruler inside the bucket. 3 Put the bucket under the tap and record the height of water every 10 seconds (You may wish to record every 30 s or so dependent upon how much water is flowing from the tap). 4 Draw a graph. 30 25 20 15 10 5 0 0 1 2 3 4 5 6 You need: • a bucket • a ruler • a stopwatch Two statisticians were flying from Perth to Sydney. About an hour into the flight, the pilot announced, 'Unfortunately, we have lost an engine, but don't worry: There are three engines left. However, instead of five hours, it will take seven hours to get to Sydney.' A little later, the pilot told the passengers that a second engine had failed. 'But we still have two engines left. We're still fine, except now it will take ten hours to get to Sydney.' Somewhat later, the pilot again came on the intercom and announced that a third engine had died. 'But never fear, because this plane can fly on a single engine. Of course, it will now take 18 hours to get to Sydney.' At this point, one statistician turned to another and said, 'Gee, I hope we don't lose that last engine, or we'll be up here forever!' Chapter 7 Solving Equations 99 Technology Technology 7.1 The Quadratic Formula and the Calculator You can be more efficient and accurate in the use of your calculator. Example: Solve: 5x2 − 2x − 4 = 0 1 Write the quadratic 2 Write the general quadratic 3 Decide values of a, b, c 5x2 − 2x − 4 = 0 ax2 + bx + c = 0 a=5 b=ˉ2 c=ˉ4 4 Write the quadratic formula x= −b ± b 2 − 4ac 2a 5 Substitute for a, b, and c x= −− 2 ± ( − 2) 2 − 4 × 5 × − 4 2×5 6 Calculate x= 2 ± 9.17 10 x= (2 + 9.17) 10 Use these brackets on your calculator. or x = (2 + 9.17) 10 x = 1.12 or x = ˉ0.72 To calculate this in one go. ( − 2) 2 − 4 × 5 × − 4 you need an extra set of brackets. ( ( − 2) 2 − 4 × 5 × − 4 ) √ ( ( ± 2 ) x2 − 4 You may need to practice this on a calculator. × 5 × ± 4 ) = to give the answer 9.17 Technology 7.2 Quadratics and the Graphics Calculator Use a graphics calculator to check your answers to Exercises 7.4, 7.5, 7.6, and 7.7. Example: Graph: y = x2 + x − 6 and solve: x2 + x − 6 = 0 Press Y= and enter the function eg., x2 + x − 6 Press Graph to see a graph of the function. To solve: x2 + x − 6 = 0 is to find the x-intercepts. a)Use CALC to find the intercepts (some calculators use zero and value). b)Use TRACE and move the cursor to the x-intercepts. c)Use 100 TABLE to find the x-intercept (where y = 0). Can you get the answers: x = ˉ3 and x = 2 Chapter Review 1 Exercise 7.13 1 2 3 4 5 6 7 x 2x − − = 2 2 3 3x − 4x= ˉ12 ˉx= ˉ12 x = 12 {×6} Check: (ˉ13+3)/5 = ˉ10/5 = ˉ2 3(x − 4) = 15 ˉ2(x − 3) = 8 7(2x − 1) = 14 2(x – 3) + 3x = 11 5(x – 3) – 2x = 12 6x − 8 = 2x + 4 2x + 2 = 16 − 5x 8 x + 5 = −1 3 9 2x −1 =2 3 10 3x x − =7 2 3 11 The sum of the interior angles of a polygon is given by the formula: S = 90(2n − 4), where n is the number of sides on the polygon. How many sides in a polygon with an interior angle sum of 540°? 12 The perimeter of a block of land is 110 m. The length is 21 m longer than the breadth. Find the length and the breadth. x2 − 8x + 16 = 0 (x − 4)2= 0 x − 4 = 0 x = 4 {perfect square} {square root both sides} {inverse of −4} Check: (4)2 − 8×4 + 16 = 0 x2 − 3x − 10 = 0 x − 5x + 2x − 10 = 0 {grouping pairs} x(x − 5) + 2(x − 5)= 0 {factorising } (x − 5)(x + 2) = 0 {factorising} Either x − 5 = 0 or x + 2 = 0 x = 5 or x = ˉ2 2 Check: (5) − 4×5 − 10 = 0 Check: (ˉ2)2 − 4×ˉ2 − 10 = 0 2 25x2 + 5x + 2 = 0 26 3x + 7x − 3 = 0 Given the quadratic: 2 27x2 – 11x + 3 = 0 28 ˉ4x2 – 3x + 1 = 0 29 The solution is: 13x2 + 4x + 4 = 0 14x2 + 10x + 25 = 0 15x2 − 2x + 1 = 0 16x2 − 8x + 16 = 0 17x2 + 5x + 4 = 0 18x2 + 8x + 12 = 0 19x2 + 5x − 6 = 0 20x2 − 2x − 8 = 0 21x2 − 5x − 6 = 0 22x2 − 4x − 12 = 0 23x2 − 10x + 9 = 0 24x2 − 8x + 12 = 0 ax2 + bx + c = 0 x= −b ± b 2 − 4ac 2a A stone is thrown vertically with a velocity of 40 m/s. The motion of the stone is described by the relationship: 5t2 - 40t + h = 0, where h is the height in metres and t is the time in seconds. How long will it take the stone to reach a height of 10 m? 30 Calculate x in the following figure: 7 x+1 x Chapter 7 Solving Equations 101 Chapter Review 2 Exercise 7.14 x 2x − − = 2 2 3 3x − 4x= ˉ12 ˉx= ˉ12 x = 12 {×6} Check: (ˉ13+3)/5 = ˉ10/5 = ˉ2 1 2 3 4 5 6 7 2(x − 3) = 10 ˉ5(x − 4) = 10 4(3x − 1) = 20 3(x – 1) + 2x = 12 7(x – 2) – 3x = 4 7x − 7 = 2x + 3 2x + 2 = 13 − 2x 8 x − 2 = −5 5 9 3x − 2 =5 2 10 2x x − =5 3 4 11 The volume, V, of a square based prism is given by the formula: V = w2h, where w is the width of the base and h is the height of the prism. If the width of the prism is 6 cm and the volume is 162 cm3, what is the height of the prism? 12 An isosceles triangle has two angles each twice the size of the third angle. What is the size of each angle? x2 − 8x + 16 = 0 (x − 4)2= 0 x − 4 = 0 x = 4 {perfect square} {square root both sides} {inverse of −4} Check: (4)2 − 8×4 + 16 = 0 x2 − 3x − 10 = 0 x − 5x + 2x − 10 = 0 {grouping pairs} x(x − 5) + 2(x − 5)= 0 {factorising } (x − 5)(x + 2) = 0 {factorising} Either x − 5 = 0 or x + 2 = 0 x = 5 or x = ˉ2 2 Check: (5) − 4×5 − 10 = 0 Check: (ˉ2)2 − 4×ˉ2 − 10 = 0 2 25x2 + 3x + 1 = 0 26 3x + 5x − 2 = 0 Given the quadratic: 2 27x2 – 7x + 5 = 0 28 ˉ2x2 – x + 1 = 0 29 ax2 + bx + c = 0 x= −b ± b 2 − 4ac 2a A stone is thrown vertically with a velocity of 50 m/s. The motion of the stone is described by the relationship: 5t2 - 50t + h = 0, where h is the height in metres and t is the time in seconds. How long will it take the stone to reach a height of 40 m? 30 Calculate x in the following figure: 102 The solution is: 13x2 + 6x + 9 = 0 14x2 + 8x + 16 = 0 15x2 − 4x + 4 = 0 16x2 − 10x + 25 = 0 17x2 + 4x + 3 = 0 18x2 + 6x + 8 = 0 19x2 + x − 6 = 0 20x2 − 3x − 10 = 0 21x2 − 5x − 6 = 0 22x2 − x − 12 = 0 23x2 − 9x + 8 = 0 24x2 − 13x + 12 = 0 x+1 Area=4 x