2 Quadratic. equations. Chapter Contents. Learning Outcomes. ... I just hope it s easy! x 2 8x + 7 = 0 (x 7)(x 1) = 0 x 7 = 0 or x 1 = 0 x = 7 or 1

Transcription

1 Quadratic Equations... I just hope it s easy! = 0 ( 7)( ) = 0 7 = 0 or = 0 = 7 or Chapter Contents :0 Solution using factors PAS5 :0 Solution by completing the square PAS5 :0 The quadratic formula PAS5 Investigation: How many solutions? :0 Choosing the best method PAS5 Fun Spot: What is an Italian referee? :05 Problems involving quadratic equations PAS5 Investigation: Temperature and altitude Fun Spot: Did you know that =? Maths Terms, Diagnostic Test, Revision Assignment, Working Mathematically Learning Outcomes PAS5 Solves linear, quadratic and simultaneous equations, solves and graphs inequalities, and rearranges literal equations. Working Mathematically Stage 5. Questioning, Applying Strategies, Communicating, Reasoning, 5 Reflecting 8

2 :0 Solution Using Factors Outcome PAS5 Prep Quiz :0 Factorise: Solve: 7 + = 0 8 = = = 0 A quadratic equation is one in which the highest power of the unknown pronumeral is. So equations such as: + + = 0, + 5 = 0, 5 = 0 and + 7 = 0 are all quadratic equations. The solving of a quadratic equation depends on the following observation (called the Null Factor Law). A quadratic equation is an equation of the second degree. If ab = 0, then at least one of a and b must be zero. Worked eamples Solve the quadratic equations: a ( )( + 7) = 0 b ( + ) = 0 c ( )( + 5) = 0 a + + = 0 b 9 = 0 c = 0 a + = b 5 = c 6 = Solutions a If ( )( + 7) = 0 b If ( + ) = 0 c If ( )( + 5) = 0 then either then either then either = 0 or + 7 = 0 = 0 or + = 0 = 0 or + 5 = 0 = or = 7 = 0 or = = or = 5 A quadratic equation can have two solutions. = -- or = To solve these equations, they are factorised first so they look like the equations in eample. a + + = 0 b 9 = 0 or 9 = 0 ( + )( + ) = 0 ( 7)( + 7) = 0 So = 9 So + = 0 So 7 = 0 = 7 or 7 or + = 0 or + 7 = 0 ie = ±7 = or = 7 or 7 To factorise an c = 0 ( )( + 5) = 0 epression like 9 5, So = 0 or + 5 = 0 = -- or 5 you can use the CROSS METHOD continued Chapter Quadratic Equations 9

3 Before these equations are solved, all the terms are gathered to one side of the equation, letting the other side be zero. a + = b 5 = c 6 = = 0 5 = = 0 ( + )( ) = 0 (5 ) = 0 ( + )( ) = 0 So + = 0 So = 0 So + = 0 or = 0 or 5 = 0 or = 0 = or = 0 or -- = -- or To solve a quadratic equation: gather all the terms to one side of the equation factorise solve the two resulting simple equations. Of course, you can always check your solutions by substitution. For eample a above: Substituting = Substituting = + = + = L.H.S. = ( ) + ( ) L.H.S. = () + () = 6 = 9 + = = = R.H.S. = R.H.S. Both = and = are solutions L.H.S. = left-hand side R.H.S. = right-hand side Eercise :0 Find the two solutions for each equation. Check by substitution to ensure your answers are correct. Foundation Worksheet :0 Quadratic equations PAS5 Factorise a b + + Solve a ( ) = 0 b ( )( + ) = 0 a ( 5) = 0 b ( + 7) = 0 c ( + ) = 0 d 5a(a ) = 0 e q(q + 5) = 0 f 6p(p 7) = 0 g ( )( ) = 0 h ( 7)( ) = 0 i (a 5)(a ) = 0 j (y + )(y + ) = 0 k (t + )(t + ) = 0 l ( + 9)( + 5) = 0 m (a 6)(a + 6) = 0 n (y + 8)(y 7) = 0 o (n + )(n ) = 0 p (a + )(a ) = 0 q ( + )( 5) = 0 r ( ) = 0 s ( )( + ) = 0 t (a )(a ) = 0 u (6y 5)(y + ) = 0 v 6(5 ) = 0 w (9y + )(7y + ) = 0 (5 )(5 + ) = 0 After factorising the left-hand side of each equation, solve the following. a + = 0 b m 5m = 0 c y + y = 0 d 6 + = 0 e 9n n = 0 f + 8 = 0 g = 0 h a 9 = 0 i y 6 = 0 j a = 0 k n 00 = 0 l m 6 = 0 m + + = 0 n a 5a + 6 = 0 o y + y + 5 = 0 p a 0a + = 0 q = 0 r m m + = 0 s h + h 0 = 0 t + 5 = 0 u a a 5 = 0 v + 56 = 0 w y 8y + 7 = 0 a + 9a 0 = 0 0 New Signpost Mathematics Enhanced

4 Factorise and solve the following. a + = 0 b = 0 c = 0 d + = 0 e 0 = 0 f = 0 g = 0 h 9 5 = 0 i + 5 = 0 j = 0 k + = 0 l = 0 m = 0 n 6 = 0 o = 0 p = 0 q 7 + = 0 r 0 + = 0 Gather all the terms to one side of each equation and then solve. a = b m = 8m c = 5 d = 5 e a = a + 5 f y = y g m = 9m 8 h n = 7n + 8 i h = h + j + = k y + y = l 7 = 0 m y + y = 8 n t + t = 8 o y + y = 5 p + = q = 5 r m m = 6 s = t 5p = 7p 6 u = 5 These are harder to factorise! Check answers by substitution. :0 Solution by Completing Outcome PAS5 the Square This method depends upon completing an algebraic epression to form a perfect square, that is, an epression of the form ( + a) or ( a). Worked eamples What must be added to the following to make perfect squares? Solutions Because ( + a) = + a + a, the coefficient of the term must be halved to give the value of a Half of 8 is, so the perfect square is: Half of 5 is --, so the perfect square is: = ( + ) ( ) 5 -- = ( -- ) Now, to solve a quadratic equation using this technique, we follow the steps in the eample below. + = 0 + = + + = + ( + ) = 5 + = ± 5 = ± 5 = or 7 Move the constant to the R.H.S. Add (half of coefficient) to both sides. Chapter Quadratic Equations

5 Note that the previous eample could have been factorised to give ( )( + 7) = 0, which, of course, is an easier and quicker way to find the solution. The method of completing the square, however, can determine the solution of quadratic equations that cannot be factorised. This can be seen in the eamples below. Worked eamples Solve: = 0 5 = 0 = 0 Solutions = = = + ( + ) = 8 + = ± 8 = ± 8 ie = + or ( 0 7 or 5 8) Note that the solution involves a square root, ie the solution is irrational. Using your calculator, approimations may be found. 5 = 0 = 5 + ( ) -- = 5 + ( -- ) ( ) -- = = ± = -- ± ie = or ( 9 or 9) = = 0 -- = -- + ( ) = -- + ( -- ) ( -- ) = 7 -- = ± = -- ± ie = or ( 55 or 0 ) When the coefficient of is not, we first of all divide each term by that coefficient. Note: You can use the following fact to check your answers. For the equation: a + b + c = 0 b the two solutions must add up to equal -- a 6 In eample, ( 0 7) + ( 5 8) = 6 [or ] In eample, 55 + ( 0 ) = [ -- ] New Signpost Mathematics Enhanced

6 Eercise :0 What number must be inserted to complete the square? a = ( +...) b = ( +...) c +... = (...) d +... = (...) e = ( +...) f = (...) g = ( +...) h +... = (...) i = ( +...) j = (...) Solve the following equations, leaving your answers in surd form. a ( ) = b ( + ) = c ( + 5) = 5 d ( ) = 0 e ( ) = 7 f ( + ) = g ( + ) = 8 h ( + 0) = i ( ) = 8 j ( + -- ) = 5 k ( -- ) = l ( + -- ) = m ( ) = -- n ( + ) = -- o ( -- ) = Solve the following equations by completing the square. Also find approimations for your answers, correct to two decimal places. a + = 0 b 5 = 0 c 8 = 0 d = 0 e 6 + = 0 f + + = 0 g + 0 = 5 h + = i = j = 0 k + 7 = 0 l + = 0 m = 0 n + 5 = 0 o + 5 = 0 p = q + = r 5 = s = 0 t + = 0 u 8 + = 0 v + = 0 w 5 = 0 = Completing the square :0 The Quadratic Formula Outcome PAS5 As we have seen in the previous section, a quadratic equation is one involving a squared term. In fact, any quadratic equation can be represented by the general form of a quadratic equation: a + b + c = 0 where a, b, c are all integers, and a is not equal to zero. If any quadratic equation is arranged in this form, a formula using the values of a, b and c can be used to find the solutions. The quadratic formula for a + b + c = 0 is: b ± b ac = a Chapter Quadratic Equations

7 This formula is very useful if you can t factorise an epression. PROOF OF THE QUADRATIC FORMULA a + b + c = 0 b -- c = 0 a a b + -- c = -- a a + b -- = a + b a b a -- c a b b = ac a a b ± b = ac a a b ± b = ac a NOTE: This proof uses the method of completing the square. Worked eamples Solve the following by using the quadratic formula = = 0 = = 0 Solutions For the equation = 0, For = 0, a =, b = 9, c =. a =, b = 5, c =. Substituting these values into the Substituting into the formula gives: formula: b ± b = ac b ± b = ac a a 5 ± 5 = ± 9 = ± 5 = ± 8 = ± = ± 9 = Since there is no rational equivalent to 9 ± 7 = the answer may be left as: = -- or = -- or = or Approimations for these answers may be found using a calculator. In this case they would be given as: 0 or 79 (to dec. pl.) New Signpost Mathematics Enhanced

8 The equation = + must first be written in the form a + b + c = 0, ie = 0 So a =, b =, c =. Substituting these values gives: b ± b = ac a ( ) ± ( ) = ( ) ± + = ± 8 = So = or (ie or 0 55 to dec. pl.) For = 0, a =, b =, c = 7. Substituting these values gives: ± = ± 5 = But 5 is not real! So = 0 has no real solutions. You should learn this formula! The solutions of the equation a + b + c = 0 are given by: b ± b ac = a Eercise :0 Foundation Worksheet :0 The quadratic formula PAS5 Use the quadratic formula to solve the following equations. All have rational answers. a = 0 b = 0 c = 0 d 0 = 0 e 5 = 0 f + = 0 b ± b Evaluate ac if: a a a =, b =, c = b a =, b = 5, c = Solve: a = 0 b = 0 g 9 + = 0 h 8 + = 0 i = 0 j = 0 k = 0 l = 0 m 5 = 0 n 5 9 = 0 o 5 + = 0 p = 0 q = 0 r 8 + = 0 Solve the following, leaving your answers in surd form. (Remember: A surd is an epression involving a square root.) a + + = 0 b + + = 0 c = 0 d + = 0 e + = 0 f + = 0 g = 0 h 7 + = 0 i 6 + = 0 j 0 9 = 0 k 8 + = 0 l = 0 m = 0 n + = 0 o 7 + = 0 p = 0 q 9 + = 0 r 5 + = 0 s + = 0 t = 0 u = 0 v + 5 = 0 w = = 0 Chapter Quadratic Equations 5

9 Use the formula to solve the following and give the answers as decimal approimations correct to two decimal places. a + = 0 b 6 + = 0 c = 0 d = 0 e + 5 = 0 f + = 0 g + = 0 h 7 = i = 6 j + = 0 k 5 = 0 l = 0 m = 7 n 5 = o 6 = + Investigation :0 How many solutions? Consider these three quadratic equations: A = 0 B = 0 C = 0 If we use the formula to solve each of these, we get: 6 ± 6 A = 5 6 ± B = ± 6 6 ± 0 = = ± = = -- = or 5 = 6 ± 6 C = ± = [ has no real solution] has no real solutions Looking at these equations, it appears that a quadratic equation may have two, one or no solutions. The key is the part of the formula under the square root sign. The number of solutions is determined by b ac. If b ac is positive then the equation will have solutions zero then the equation will have solution negative then the equation will have no solution. Eercises By evaluating b ac for each equation, determine how many solutions it will have. + + = = = 0 = 0 5 = = 0 b ac is called the discriminant = = = = = = 0 6 New Signpost Mathematics Enhanced

10 :0 Choosing the Outcome PAS5 Best Method Prep Quiz :0 Factorise: Solve: 5 ( )( + 7) = 0 6 ( )( + ) = = 0 8 = = 0 0 Write down the formula for the solution of the equation: a + b + c = 0. Some quadratic equations may appear in a different form from those we have seen so far, but they can always be simplified to the general form a + b + c = 0. They may then be factorised, or the formula applied, to solve them. Worked eamples olve the following equations. + = + 6 ( 5) = 6 Solutions In this eample, all the terms must be gathered to the L.H.S. + = = = 0 This cannot be factorised, so the quadratic formula is used. b ± b = ac a 5 ± = ± 5 = = or L.H.S. left-handside 5 6 = Epand and gather the terms to the L.H.S. ( 5) = 6 5 = = 0 Factorising gives: ( 6)( + ) = 0 = 6 or 5 6 = Multiplying both sides by gives: = 5 6 ie = 0 Factorising gives: ( )( ) = 0 = or When solving a quadratic equation: Step : Epress the equation in general form a + b + c = 0 Step : Factorise, if you can, and solve it OR Use the formula = b ± b ac a Factorise if you can! Chapter Quadratic Equations 7

11 Worked eample This is an eception to the above rule! (a + 7) = 6 For equations like this, where one side is a perfect square, it is easier to follow the final steps in the method of completing the square. Solution (a + 7) = 6 a + 7 = ± 6 a = 7 ± a = or 7 6 Eercise :0 Solve the following quadratic equations. (Give irrational answers to dec. pl. if necessary.) a = 0 b 8 + = 0 c + 5 = 0 d + = 0 e + = 0 f + + = 0 g + 8 = 0 h 0 = 0 i 5 0 = 0 j 8 = 0 k = 0 l 9 = 0 m + + = 0 n = 0 o 5 + = 0 p = 0 q = 0 r 6 8 = 0 s = t = 6 u 5 = v 5 = 0 w 6 = = 9 5 Rearrange each equation below to the form a + b + c = 0, and solve. a + 9 = b + 0 = c + 0 = + d + 5 = 6 e + 5 = f + 0 = g + 5 = + h + 7 = 5 i + = + j ( + 5) = 6 k ( 7) = 8 l = ( + 8) m (m ) = n ( + ) = 9 o ( + 5) = p (a + ) = 6 q (5y ) = 7 r (6n 7) = s = t = u = v w ( + ) = = ( + ) = -- 8 New Signpost Mathematics Enhanced

12 Fun Spot :0 What is an Italian referee? Work out the answer to each question and put the letter for that part in the bo that is above the correct answer. Solve: A ( + )( 5) = 0 M ( + )( ) = 0 R ( )( 7) = 0 R ( + 5)( ) = 0 Solve by factorising: M = 0 A 6 = 0 E 5 6 = 0 I + 6 = 0 Solve: N ( ) = 5 P ( + ) = O ( + 7) = 9 U ( + ) = I came, I saw and you re offside!, --, 7, 0,, 5 ± 5 ±, ±, 5 --, --, 6 :05 Problems Involving Outcome PAS5 Quadratic Equations Prep Quiz :05 From the list of numbers,,, --, 5, 9, 0, write down the numbers which are: integers odd square Write down the net consecutive integers after: 8 5 n Write down the net consecutive even numbers after: ( is even) Write down epressions for: 8 a number that is less than a cm 9 the area of this rectangle 0 the perimeter of this rectangle (a ) cm Sometimes, when solving a problem or applying a given formula, a quadratic equation may be involved. Consider the following eamples. Worked eamples The product of two consecutive positive even numbers is 8. Find the numbers. (Hint: If the first number is, then the net even number is + ). The length of a rectangle is 5 cm longer than its breadth. If the area of the rectangle is 8 cm, find the length of the rectangle. A projectile is fired vertically upwards and its height h in metres after t seconds is given by the formula: h = 0t 8t Find the time taken by the projectile to first reach a height of 8 metres. Chapter Quadratic Equations 9

13 Solutions The problem gives the equation: ( + ) = 8 Solving this gives: + = = 0 ( + 8)( 6) = 0 ie = 8 or 6 Since the numbers are positive, must equal 6. The two consecutive integers are 6 and 8. If the breadth is, then the length is Since the area of a rectangle is equal to length times breadth, then: ( + 5) = = = 0 ( + )( 7) = 0 So = or 7 Now, since the dimensions must be positive, the breadth must be 7 cm. The length = cm. h = 0t 8t To find the time t when the height h = 8 metres, substitute into the formula: so 8 = 0t 8t Gathering all the terms on the L.H.S. 8t 0t + 8 = 0 8(t 5t + 6) = 0 8(t )(t ) = 0 t = or The projectile was at a height of 8 m after seconds and seconds. So it was first at a height of 8 m after second A projectile can be at the same height at two different times! Eercise :05 Find the two positive integers required, if: a the numbers are consecutive and their product is 0 b the numbers are consecutive and their product is 90 c the numbers are consecutive even numbers and their product is 0 d the numbers are consecutive odd numbers and their product is 6 Consecutive means one after the other. 0 New Signpost Mathematics Enhanced

14 a b c d e a The sum of a positive integer and its square is 90. Find the number. The sum of a positive integer and its square is. Find the number. The difference between a positive integer and its square is 56. Find the number. The square of a number is equal to 5 times the number. What are the two possible answers? When a number is subtracted from its square, the result is. Find the two possible solutions. Find the dimensions of this rectangle if the length is 6 cm longer than the breadth and its area is 0 cm. b + 6 The width of a rectangular room is metres shorter than its length. If the area of the room is 55 m, find the dimensions of the room. c The base of a triangle is 5 cm longer than its height. If the area of the triangle is 7 cm, find the length of the base. 5 d A right-angled triangle is drawn so that the hypotenuse is twice the shortest side plus cm, and the other side is twice the shortest side less cm. Find the length of the hypotenuse. 5 6 a Michelle threw a ball vertically upwards, with its height h, in metres, after a time of t seconds, being given by the formula: h = 8t t Find after what time the ball is first at a height of m. b The sum, S, of the first n positive integers is given by the formula n S = -- ( n + ) Find the number of positive integers needed to give a total of 78. c For the formula s = ut + -- at, find the values of t if: i s = 8, u = 7, a = ii s = 6, u =, a = iii s = 7, u =, a = 6 An n-sided polygon has -- n(n ) diagonals. How many sides has a figure if it has 90 diagonals? Jenny is y years old and her daughter Allyson is y years old. If Jenny lives to the age of y, Allyson will be y years old. How old is Allyson now? (Note: the difference in ages must remain constant.) Chapter Quadratic Equations

15 7 8 9 Kylie bought an item for $ and sold it for $0.56. If Kylie incurred a loss of per cent, find. A relationship that is used to approimate car stopping distances (d) in ideal road and weather conditions is: d = t r v + kv where t r is the driver s reaction time, v is the velocity and k is a constant. a Stirling s reaction time was measured to be 0 8 seconds. The distance it took him to stop while travelling at 0 m/s (7 km/h) was 5 metres. Substitute this information into the formula to find the value of k. b If, for these particular conditions, Stirling s breaking distance 80 d is given by 70 d = 0 8v v 60 complete the table below, finding d correct to the nearest metre 50 in each case. 0 v (in m/s) d (in m) 0 0 c Graph d against v using the number plane shown on the right v What kind of curve is produced? d Use your graph to find the velocity (in m/s) that would produce a stopping distance of 0 metres. Check your accuracy by solving the equation 0 = 0 8v v using the formula v = b + b ac a e What factors would determine the safe car separation distance in traffic? The rise and tread of a staircase have been connected using rise the formula r = -- ( t), or r = 66/t where r and t are measured in inches. (One inch is about 5 centimetres.) a If the tread should not be less than 9 inches, what can be said about the rise? tread b Graph both functions on the same set of aes and compare the information they provide. c What are the points of intersection of the two graphs? Check the accuracy of your graphs by solving the two simultaneous equations, r = -- ( t)... r = 66/t... (Hint: Substitute into and solve the resulting quadratic equation.) d Convert formulae and to formulae applicable to centimetres rather than inches. e Do the measurements of staircases you have eperienced fit these formulae? Investigate other methods used by builders to determine r and t. New Signpost Mathematics Enhanced

16 Investigation :05 Temperature and altitude The following formula has been used to give the boiling point of water at various heights above sea-level: h = 50( T) + ( T) where height, h, is in feet and the temperature, T, is in degrees Fahrenheit ( F). Show that h = ( T)(7 T). At what height above sea-level does water boil at: a 00 F? b 50 F? Plot a graph of h against T. (Use values of T from 60 F to 80 F. Use values of h from feet to feet.) T in F h in feet Use your graph to find the temperature at which water boils: a at Flinders Peak (55 feet above sea-level) b atop Mt Everest (8 000 feet above sea-level) c at the bottom of the Mindanao trench (5 000 feet below sea-level). 5 Check your answers to question by substituting each height into the formula and solving the resulting quadratic equation. 6 Change the units on the aes of your graph so that they are in degrees Celsius ( C) and thousands of metres. To do this, use the 5 formula C = -- (F ) and the approimation, 9 foot = 0 05 metres. 7 Discuss: Over what temperature range is this formula useful or valid? Find the height at which the space shuttle orbits. Can the formula be used there? Is the shuttle pressurised to some equivalent height above sea-level? Thousands of feet h Degrees F T Chapter Quadratic Equations

17 Fun Spot :05 Did you know that =? Now that your algebra skills are more developed, you can be let into the secret that really is equal to. Proof: Assume that = y. Multiply both sides by, = y Subtract y from both sides, y = y y Factorise both sides, ( y)( + y) = y( y) Divide by ( y), + y = y Now, if we let = y =, then = Q.E.D. Pretty clever eh! Of course there is a fallacy in the proof above. Can you find it? Literacy in Maths Maths terms coefficient The number that multiplies a pronumeral in an equation or algebraic epression. eg + 5 = 0 coefficient of is coefficient of is completing the square Completing an algebraic epression to form a perfect square, ie ( + a) or ( a). eg To complete the square for + 6, the number 9 is added, thus: = ( + ) factorise To write an epression as the product of its factors. The reverse of epanding. quadratic equation An equation in which the highest power of the unknown pronumeral is. eg 6 = 0, = 0 A quadratic equation may have two solutions. quadratic formula A formula that gives the solutions to equations of the form a + b + c = 0. = b ± b ac a Maths terms New Signpost Mathematics Enhanced