SOLUTION OF QUADRATIC EQUATIONS

Transcription

1 SOLUTION OF QUADRATIC EQUATIONS * The standard form of a quadratic equation is a + b + c = 0 * The solutions to an equation are called the roots of the equation. * There are methods used to solve these equations : A. by FACTORISATION B. by COMPLETING THE SQUARE C. by using the QUADRATIC FORMULA b b ac a SOLVING QUADRATIC EQUATIONS BY FACTORISATION * If ab 0, then a 0 and / or b 0 e.g. if ( )( ) 0, then 0 or 0 or 0 0 or or. Standard form or with brackets Eamples : Solve for : a) 0 (std form) b) (std form) ( ) 0 0 ( ) 0 or ( )( ) 0 or c) ( )( ) d) ( 6) ( 5) (std form) 7 0 (std form) ( 6)( ) 0 ( 8)( 9) 0 6 or 8 or 9. Ensure equation is in standard form i.e. a b c 0. If not, remove brackets (if any), transpose and add like terms first.. Factorise.. Let each factor = 0.. Solve for the unknown.

2 . Removing fractions Eamples : Solve for : a) c) b) L.C.D.= ; 0 L.C.D.= ( )( ) ; ( ) ( ) ( ) or 8 0 (std form) 9 0 (std form) ( )( ) 0 ( )( ) 0 or or 6 6 ( )( ) ( )( ) L.C.D.= ( )( )( ); ; ; ( )( ) ( 6)( ) ( ) ( ) (std form) ( )( ) 0 or invalid. Factorise the denominator if necessary.. Find the L.C.D. state restrictions.. Multiply throughout by the L.C.D.. Bring the equation to standard form by removing brackets,adding like terms and transposing. 5. Factorise. 6. Solve for the unknown, rejecting any invalid answers.

3 . Using substitution ( k-method) Eamples : Solve for : a) 9 b) 0 let k let k 9 k k 0 k k L.C.D.= k ; k 0 L.C.D.= k ; k 0 k 9 k k 0 k ( k )( k ) 0 k or k SO 6 0 (std form) 0 (std form) ( )( ) 0 ( )( ) 0 or or OR 0 (std form) 0 (std form) ( ) 0 ( ) 0 0 or. Let what is common equal k.. Substitute k into the original equation.. Find a L.C.D. if necessary.. Get the equation in standard form. 5. Factorise. 6. Solve for k. 7. Substitute k -values into let statement. 8. Get the equation in standard form. 9. Factorise. 0. Solve for the unknown.. Check that all solutions are valid

4 . Surd equations Eamples : Solve for : a) 5 6 b) (std form) 9 0 ( 6)( ) (std form) 6 or 0 ( )( ) invalid or invalid. Isolate the square root.. Remove the square root, by squaring both sides of the equation.. Write the equation in standard form.. Factorise. 5. Solve for the unknown. 6. Check the solutions, by substituting into the original equation. 5. Finding square roots Eamples : Solve for : a) 5 b) ( ) 9 5 9, or 5 or c) 5 0 d) ( ) 5 ( ) 5 or no solution. Isolate the square term.. Square root both sides of the equation.. Solve for the unknown.

5 SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE Pre-knowledge : The Perfect Square Trinomial. Find the products of the following :. ( ). ( 9)... Factorise the following : Find the value of k and hence find the perfect square for each of the following : 6 k k.... k Conclusion : 5 k k k k k The third term k can be found by using the following formula : k Method for finding the perfect square : st term ; sign of nd term ; rd term 5

6 N.B. Only use this method of solving when asked to. Eamples : Solve for, by completing the square : a) 8 7 b) 0 8 () 7 ( ) 9 () 7 or c) (divide by ) or add ( of co-efficient of ) 6 6 to both sides 5 7 factorise perfect square trinomial on LHS square root both sides (answer in surd form) 6 6 0,59 or, 6 (answer to d.p.). Ensure that a. If not, divide all terms by a.. Transpose c.. Complete the square and balance the equation by adding. Find the perfect square. 5. Square root both sides of the equation. 6. Solve for the unknown. b. 6

7 SOLVING EQUATIONS BY USING THE QUADRATIC FORMULA Deriving the formula by completing the square : a b c 0 b b a a b a c a b b ac a a b a b ac a b b ac a N.B. the formula is used when you cannot factorise. Eamples : Solve for : a) 5 0 (std form) b) 5 a ; b 5; c 5 0 (std form) a ; b 5; c b (5) b ac a (5) () ()( ) b (5) b ac a ,9 or, 69 ( d.p.) no real solution (5) (). Write the equation in standard form.. Find a, b and c.. Write down the quadratic formula.. Substitute for a, b and c in the formula. 5. Solve for the unknown (answers in simplest surd form or rounded) ()() 7

8 SIMULTANEOUS EQUATIONS Simultaneous equations in two variables consist of two equations, generally containing both variables. When solving these equations, we find values for the two variables that will satisfy both equations simultaneously. Graphically, the solutions are the points of intersection of the two graphs of the functions. Eample : Solve for and y if y 5 and y y 8 y 5...() y y 8...() From () : 5 y...() Substitute () into () : y y(5 y) (5 y) 8 y y 5y y 5y y (5 0y y 00 80y 6y ) 8 8 y 75y 08 0 (divide by -) y 5y 6 0 (std form) ( y 9)( y ) 0 9 y or y 9 Substitute into () : 5 or 5 () Points of intersection : 9 ; and ( ;) See also eample on page 80. Make one of the unknowns of the linear equation the subject of the formula. Avoid fractions where possible.. Substitute this into the other equation (usually quadratic).. Write the quadratic equation in standard form.. Solve for the unknown. 5. Substitute these values into the linear equation to find the corresponding values of the other unknown. 8

9 FINDING A QUADRATIC EQUATION FROM THE ROOTS The roots of the equation are the solutions i.e. -values Type Given one root and asked to find an unknown and the other root. Eample : If one root of an equation k 0 is -, find k and the other root. Substitute : Substitute k : ( ) ( ) k 0 0 k 0 ( )( ) 0 k or the other root is Type Given the roots of an equation and asked to find the equation a b c 0. Eamples : Find the equation whose roots are : a) - and b) and -5 and and 5 ( )( ) 0 ( )( 5) N.B. If then and 0 9

10 PROBLEMS LEADING TO QUADRATIC EQUATIONS Method ;. Read the problem carefully to decide what you want to determine and choose variables to represent the unknown quantities.. Epress the relationship of the quantities in equations.. Solve the equations.. Interpret your answers and reject the unrealistic roots. Eamples : a) Sara s father is si times as old as Sara. The product of their ages is 50 years. What are their respective ages? Let Sara s age be years and her father s age 6 years. (6) Sara is 5 years old and her father is 0 years old. (reject -5) b) A man travels 80 km from his farm to town in a loaded truck. On the return journey he is able to travel 0 km/h faster having unloaded his produce in town. He saves hour on the return journey by travelling faster. At what speed did he travel to town? Journey Distance Speed Time From farm to town 80km 80 From town to farm 80km N.B. the time taken on the trip to town eceeded the time 0 taken on the return trip by hour. L.C.D.= ( 0) ; 0; 0 80( 0) 80 ( 0) (std form) 0 ( 60)( 90) 60 or 90 he travelled at 60 km/h to town. (reject -90 0

11 Eercise. Lerato is four years older than her brother, John. If the product of their ages is 96, what are their respective ages? [Lerato yrs old ; John 8 yrs old]. Dan s tai is three years older than Eric s tai. In two year s time, the product of the ages of the two tai s will be 5. How old is each tai at present? [Dan s tai 7 yrs old ; Eric s tai yrs old]. An elephant is twice as old as a lion. Three years ago the product of their ages was 90. How old is the elephant now? [Elephant 8 yrs old]. The product of two consecutive even integers is 68. Find the integers. [ and or - and -] 5. The product of two consecutive odd integers is. Find the integers. [ and or - and -] 6. The difference between two natural numbers is and their product is 0. Calculate the numbers. [5 and 8] N.B. Distance = Speed Time Speed = Distance Time Time = Distance Speed

12 QUADRATIC INEQUALITIES Eamples : Solve for : a) 6 0 ( )( ) 0 Critical values: - and or OR ( ; ] or [ ; ) b) < 0 0 < 0 ( 5)( 8) < 0 Critical values : -5 and 8 5< < 8 OR (5;8). Write the equation in standard form.. Factorise.(could be HCF, DOTS, TRINOMIAL). Find the critical values.. Represent the information on a number line. 5. Write the solution as an inequality or using interval notation. HW Pg 6 E 8 No. and Pg N0s. 5. and 5.