CHAPTER 2- QUADRATIC EQUATIONS

Transcription

1 Chapter - Quadratic Equations CHAPTER - QUADRATIC EQUATIONS. INTRODUCTION. General form of quadratic equation is a + + c where : (i) is unknown (ii) a, and c is constant (iii) a 0 (iv)the powers of are positive integers up to a maimum value of.. Roots are the value of the unknown that satisfy the equation. Eample : 3 ( + )( 3) + or 3 = or = 3 root. SOLVING QUADRATIC EQUATIONS. Factorization method Eample : 3 ( + )( 3) + or 3 = or = 3 Eample : 3 4 ( + )( 4) + or 4 = or = 4 Page 0

2 Chapter - Quadratic Equations.Completing the square method Eample: 3 + ( ) ( ) ( ) 3 ( ) ( ) 4 = 4 = ± = or = = or = 3 Part + ( ) 3. By Using formula 3 ( ) while ( ) 3 To solve the quadratic equation y completing the square method, the coefficient of must e. If ( ) ( ), we know that the solution is equal to zero. For this eample the coefficient of is - so - is divided y and ecome -. ( is factorized and ecomes is solved. ) ( ) is added etween term and c. The concept of completing the square method is the coefficient of which is is divided y and the numer is squared If the coefficient of is -4 so -4 is divided y and ecome -. So the equation will ecome like this 4 + ( ) ( ) 3=0. If the coefficient of is 6 so 6 is divided y and ecome 3. So the equation will ecome like this (3) (3) 3=0. a + + c c + + a a + + ( ) ( ) a a a ( + ) + a 4a ( + ) = a 4a c a c(4a) a(4a) + c a How to otain the formula? To otain the formula is y using completing the square method. ( + ) a 4ac = 4a 4ac + = ± a 4a 4ac + = ± a 4a Page

3 Chapter - Quadratic Equations + a = ± 4ac a = a ± ± = Eample: 4ac a 4ac a This is the formula. We can just sustitute the value of a, and c from the equation ased on the general form a + + c to find the values of Eample: Solve the equation 3 y using quadratic formula. From the equation, we know that a =, = and c = 3. So, we can just sustitute the value into the formula, ( ) ± = ( ) ± (4 + = ± (6 = () 4()( 3) ± 4 = + 4 = or = 4 The value of ± 4is 4 and -4. So convert the equation into two where the + 4 equation = and the other one = 3 or = EXERCISE.. Find the roots of the quadratic equation = 4 y using completing the square method. Give your answer correct to decimal places.. Solve the following quadratic equation using the quadratic formula. (a) 5 3 () = Factorize the following quadratic equations and hence, state their roots. (a) 5 3 () = 7 4 Page

4 Chapter - Quadratic Equations.3 FORMING QUADRATIC EQUATION FROM THE GIVEN ROOTS Given roots are and 4, = 4 or = Sum of roots = 4 4 or + = 3 ( + )( 4) Product of roots = = The general form is ( S. + ( P. Sustitute S.R = 3 and P.R = 4 (3) + ( 4) 3 4 EXERCISE.3. Write quadratic equations with roots 3 and Form a quadratic equation whose roots are -3 and 4 3. Write quadratic equations with roots and -..4 SUM OF ROOTS (S. AND PRODUCT OF ROOTS (P. If the roots are a and, = a or = a or ( a)( ) a + a ( a + ) + a What is the general form? a and is the roots so in the equation, Hence, the general form is a + is the sum of roots and ais the product of the roots ( S. + ( P. Eample: The roots of the equation 4 + are m and n. Find the equation whose roots are 3m and 3n Make the equation in the general form ( S. + ( P. y divide all terms y. This is ecause in the general form, the value of a must e. Page 3

5 Chapter - Quadratic Equations From the equation aove, we know that S.R = P.R = Given the roots are m and n. Hence, m + n = mn = If the roots are 3m and 3n, S.R = 3m + 3n = 3(m +n) 3 General form is ( S. + ( P.. For this question, the equation is +. Compare these two equations. We know that the sum of roots of the equation is and the product of roots of the equation is. Given that m and n is the roots, so m + n = and mn = P.R= 3m. 3n = 9mn 4 Sustitute into 3 and into 4, S.R = 3() = 6 P.R = 9( ) = 9 the equation whose roots are 3m and 3n is We can leave the equation with ut it is etter to let the equation without fraction so we multiply all terms with. EXERCISE.4. Given that a and 3 are roots of the quadratic equation p , find the value of a and p.. One of the roots of the quadratic equation + p + 8 is half the value of the other root. Find the possile values of p. 3. Given that the value of one root is 3 times the other for the quadratic equation 3 + p. find (a) the value of p () the two roots Page 4

6 Chapter - Quadratic Equations.5 CONDITIONS FOR THE TYPES OF ROOT OF QUADRATIC EQUATION ± 4ac.From the formula =, we know that the part 4acis called the discriminant of a quadratic equation a + + c.. The value of the discriminate will determine the types of roots of a quadratic equation. 3. We can solve a quadratic equation y factorization if the value for 4acis a perfect square. Types of root of Quadratic Equation - If 4ac > 0, then the quadratic equation has two different roots(also known as two distinct roots) 3 4ac = ( ) 4()( 3 ) ( + )( 3) = 6 + or 3 4ac > 0 = or = 3 - If 4ac, then the quadratic equation has two equal roots ac = ( 0) 4()(5 ) ( 5)( 5) 5 4ac = 5 3- If 4ac < 0, then the quadratic equation has no real roots(or no roots) ac = ( 3) 4()(0 ) = 9 80 = - 7 4ac < 0 4- If 4ac 0, then the quadratic equation has real roots. Eample : Given that 3 and k are roots of the quadratic equation ( + ) = has two equal roots. Find the value of h. + When compare the equation ( S. + ( P. and ( ) + ( ), we would know sum of roots and product of roots for the equation. Page 5

7 Chapter - Quadratic Equations ( ) + ( ) From the equation aove, we know that S.R = P.R = Given 3 and k are roots, S.R= k + 3 P.R= 3. k = 3 k Hence, k + 3 = or 3k = k = 4 or k = 4 k = 4 From the equation, we know that S.R and P.R are and respectively. From the given roots, we know that S.R and P.R are k + 3and 3k respectively. Hence compare oth of them to find the value of k. Eample : Given that the equation 4 + k + has two different roots, find the largest integer of k. From the equation 4 + k +, we know that a =, = 4and c = k +. Two different roots: 4ac > 0 ( 4) 4()( k + ) > 0 6 4( k + ) > 0 6 4k 4 > 0 4k > k < 3 Hence, the largest integer of k is. Eample 3: Integer is a positive or negative numer including 0. k is less that 3 so the k=,,0,-,- and so on. Hence, the largest integer of k is One of the roots of the equation + k + is thrice the value of the other. Find the possile values of k. + k + Let the roots e m and 3m. From the equation, S.R = k P.R = We can choose other unknown to e the roots ut it is etter to do not put as the roots. But we cannot put k as the root. This is ecause in this case, k acts as the S.R. Page 6

8 Chapter - Quadratic Equations From the roots, S.R = 3m + m = 4m P.R = 3m m = 3m Hence, 4 m = k k = 4m 3m = m = 4 m = ± From the equation, we know that S.R and P.R are k and respectively. From the given roots, we know that S.R and P.R are 4mand 3m respectively. Hence compare oth of them to find the value of k. Sustitute m = ± into, (i) m = (ii) m = k = 4() k = 4( ) k = 8 k = 8 So, k = ± 8 Eample 4: Given that = h( ) has equal roots, find the values of h = h h 5 h h (5 + h) + (5 + h) From the equation aove, we know that a =, = ( 5 + h) and c = 5 + h. Equal roots: 4ac [ (5 + h )] 4()(5 + h) (5 + h ) 4(5 + h) h + 0h h h + 6h + 5 ( h + 5)( h + ) h + 5 or h + h = 5 or h = Page 7

9 Chapter - Quadratic Equations Eample 5: Find the largest integer value of k if k + (k 7) + k has real roots. k + (k 7) + k Real roots: 4ac 0 (k 7) 4( k)( k) 0 4k 8k k 8k k 4 The largest integer value of k is. In Form One, We have learned aout integer. Integer is a positive or negative numer that is a whole numer. Such as, and so on. Fractions and decimals are not integer. Page 8

10 Chapter - Quadratic Equations CHAPTER REVIEW EXERCISE. Solve the equation + 5 = 6.. Given and 4are roots of a quadratic equation state the equation in the form a + + c 3 where a, and c are integers. 3. Find the range of values of p of the equation ( ) = p + 5has two different roots. 4. Find the values of k such that equation ( k ) 3( k 6) + k 6 has equal roots. Hence, find the roots of the equation ased on the larger value of k. 5. Given that m + 3and n are roots of equation + 6 = 5, find the possile values of m and n. 6. The quadratic equation + m + k has roots 7and 4. Find (i) the values ok m and k (ii) the range of values of p so that + m + k = pdoes not have real roots 7. Given that equation 6 = k has different roots, find the range of values of k. 8. Given that α and β are roots of equation + k + 3 equation 7 + m. Calculate the possile values of k and m., whereas α and β are roots of 9. Given that the roots of the quadratic equation ( )( + 5) are p and q. Form a quadratic equation with roots p + and q The quadratic equation ( + 4) = p 3has two distinct roots. Find the range of values of p.. Form a quadratic equation with the roots and. 3. Given that the quadratic equation 5m + n has two equal roots. Epress n in terms of m. 3. Determine the type of roots for the quadratic equation Find the value of h if the straight line + y = kis a tangent to the curve y = 8. Page 9