qwertyuiopasdfghjklzxcvbnmqwertyui opasdfghjklzxcvbnmqwertyuiopasdfgh jklzxcvbnmqwertyuiopasdfghjklzxcvb nmqwertyuiopasdfghjklzxcvbnmqwer QUADRATIC EQUATIONS tyuiopasdfghjklzxcvbnmqwertyuiopas dfghjklzxcvbnmqwertyuiopasdfghjklzx cvbnmqwertyuiopasdfghjklzxcvbnmq wertyuiopasdfghjklzxcvbnmqwertyuio pasdfghjklzxcvbnmqwertyuiopasdfghj klzxcvbnmqwertyuiopasdfghjklzxcvbn mqwertyuiopasdfghjklzxcvbnmqwerty uiopasdfghjklzxcvbnmqwertyuiopasdf ghjklzxcvbnmqwertyuiopasdfghjklzxc vbnmqwertyuiopasdfghjklzxcvbnmrty uiopasdfghjklzxcvbnmqwertyuiopasdf ghjklzxcvbnmqwertyuiopasdfghjklzxc vbnmqwertyuiopasdfghjklzxcvbnmqw ertyuiopasdfghjklzxcvbnmqwertyuiop Name ........................................................................................ Quadratic Equations CHAPTER 2: QUADRATIC EQUATIONS 1.1 Recognize Quadratic Equations and express it in general form General form ax 2 bx + c = 0 , where a , b and c are constants , a 0 Properties 1. Equation must be in one unknown only 2. The highest power of the unknown is 2 Examples 1. 2x 2 + 3x – 1 = 0 is a quadratic equation 2. 4x 2 – 9 = 0 is a quadratic equation 3. 8x 3 – 4x2 = 0 is not a quadratic equation Activity 1 1. Determine whether each of the following equation is a quadratic equation or not . Equations Answer (a) x 2 – x = 0 Yes (b) 2x 2 – y = 0 (c) 3x + 2 = 0 (d) 2m 2 – 7m – 3 = 0 (e) k 2 – 4k = 0 (f) y 2 – 2 = 0 2. Rewrite each of the following quadratic equation in the general form. State the value of a , b and c . Quadratic equations (a) 1 + 2x = x(x + 3) 1 + 2x = x2 + 3x x2 + x – 1 = 0 (b) m 2 = 21 – 4m zefry@sas.edu.my Value of a , b and c a =1 b =1 c = -1 Quadratic Equations (c) (y + 6)(y – 2) = - 7 (d) x 2 = 7x 3 2 (e) (x + 1) 2 = 16 1.2 Roots of Quadratic Equations Notes 1. The root of a quadratic equation is the value(number) of the unknown(variable) that satisfy the equation . 2. A quadratic equation has at most two roots only Exercises 1. Determine which of the values of the variable x given are roots of the respective quadratic equation. 1 (a) x 2 – x – 2 = 0 ; x = - 1 , 1 , 2 (b) 2x 2 + 7x + 3 = 0 ; x = - 3, - , 1 , 3 2 2. Determine by inspection which of the values of x are roots of the following quadratic equations . (a) (x + 3)(x – 2) = 0 ; x = 3 , 2 , - 3 (b) x(x + 4) = 0 ; x = 4 , 0 , - 4 3. If x = 2 is the root of the quadratic equation x 2 – 3kx -10 = 0 , find the value of k . zefry@sas.edu.my Quadratic Equations 2 . SOLVING QUADRATIC EQUATION 2.1 Solving Quadratic Equations A. By Factorization If a quadratic equation can be factorized into a product of two factors such that (x – p)(x – q) = 0 , Hence x – p = 0 or x – q = 0 x = p or x=q p and q are the roots of the equation . Notes 1. If p q the equation have two different roots 2. If p = q the equation have two equal roots (one root only) 3. The equation must be written in general form ax 2 + bx + c = 0 before factorization. Activity 2 Solve the following quadratic equations by factorization . 1. x 2 – 7x – 8 = 0 2. x 2 – 4x + 4 = 0 (x–8)(x+1)=0 x – 8 = 0 or x + 1 = 0 x = 8 or x = -1 3. x 2 – 8x = 0 4. 4x 2 – 9 = 0 5. 6x 2 + 13x – 5 = 0 6. (3x + 1)(x - 1) = 7 7. 40 3x x 5 2x zefry@sas.edu.my 8. (x + 1)(x – 5) = 16 x2 – 4x – 5 = 16 x2 – 4x – 21 = 0 (x–7)(x–3)=0 x = 7 or x = 3 Quadratic Equations 9. 10. (2p + 1)(p + 1) = 0 t t 16 9t Exercise 1 Solve the following quadratic equation by factorisation. 1. x 2 – 5x – 6 = 0 [6,-1] 9. x 2 – 9x + 20 = 0 2 2. m + 5m – 24 = 0 [-8,3] 10. 4x 2 – 13x + 3 = 0 [5,4] [ 1 ,3 ] 4 2 3. y + 10y + 24 = 0 2 4. 2x + 3x – 5 = 0 [-6,-4] 2 11. 2x – 3 = 5x 2 [ 1 ,3 ] 2 [ 1 , 7 ] 2 3 [ 5,-2] 12. 6x – 11x = 7 5. 16x 2 – 6x – 7 = 0 [ 1, 5 ] 2 [ 7 , 1 ] 8 2 6. 2a 2 + 4a = 0 [0.-2] 14. (3m + 1)(m – 1) = 7 [ 2, 4 ] 7. 100 – 9n 2 = 0 [ 10 , 10 ] 15. 10x 2 + 4 = 13x [ [ 1 ,3 ] 2 16. x(x + 4) = 21 [ -7,3] 13. (2x – 3) 2 = 49 3 3 8. (2x + 1)(x + 3 ) = 0 3 B. By Completing the Square Notes 1. The expression x 2 – 2x + 1 can be written in the form (x – 1) 2 This is called “perfect square”. Example Solve each of the following quadratic equation (a) (x + 1) 2 = 9 (b) x 2 = 49 x+1=3 x + 1 = 3 , x + 1 = -3 x=2 , x=-4 (c) (x + 2) 2 = 36 zefry@sas.edu.my 1 4] , 2 5 Quadratic Equations 2. From the example , note that, if the algebraic expression on the LHS of the quadratic equation are perfect squares , the roots can be easily obtained by finding the square roots. 3. To make any quadratic expression x2 + hx into a perfect square , we add the term h ( ) 2 to the expression . 2 2 h h And this will make x hx x hx x 2 2 2 2 2 4. To solve the equation by using completing the square method for quadratic equation ax 2 + hx + k = 0 , follow this steps ; Step 1 : Rewrite the equation in the form ax 2 + hx = - k Step 2 : If the coefficient of x2 is 1 , reduce the coefficient to 1 (by dividing) . h Step 3 : Add ( )2 to both sides of the equation. 2 Step 4 : Write the expression on the LHS as perfect square. Step 5 : Solve the equation Examples 1. x2 + 6x – 9 = 0 x2 + 6x = 9 2 2. 2x2 – 5x – 8 = 0 2 6 6 x + 6x + = 9 + 2 2 2 ( x + 3 ) = 18 x+3 = 18 x+3 = 4.243 x = 4.243 – 3 , x = -4.243 – 3 x = 1.243 , x = -7.243 2 Exercise 2 Solve the following equations by completing the square. (Give your answers correct to four significant figures) 1. 2. 3. 4. 5. 6. 7. x 2 – 8x + 14 = 0 2x 2 – 7x – 1 = 0 x 2 + 5x + 1 = 0 – x 2 – 3x + 5 = 0 x 2 = 5(x + 4) -4x 2 – 12x + 3 = 0 2x 2 – 3x – 4 = 0 zefry@sas.edu.my [5.41 , 2.59] [3.64 , -0.14] [-0.209,-4.79] [-4.19,1.19] [7.62 , -2.62] [-3.23,0.232] [2.35,-0.85] Quadratic Equations C. By Using the quadratic formula The quadratic equation ax 2 + bx + c can be solved by using the quadratic formula b b 2 4ac x = , where a 0 2a Example 2x 2 – 7x – 3 = 0 a = 2 , b = -7 , c = -3 x (7) (7) 2 4(2)(3) 2(2) x 7 73 7 8.5440 4 4 x = 3.886 , -0.386 Exercise 3 Use the quadratic formula to find the solutions of the following equations. Give your answers correct to three decimal places . 1. x 2 – 3x – 5 = 0 [4.193 , -1.193] 2 2. 9x = 24x – 16 [1.333 ] 2 3. 2x + 5x – 1 = 0 [0.186 , -2.686] 4. 3x 2 + 14x – 9 = 0 [2.899 , -6.899] 2 5. 7 + 5x – x = 0 [0.768 , -0.434] 2 6. m = 20 – 4m [0.573 , -5.239] k 1 7. [-1.140 , 6.140] k2 3 8. x(x + 4) = 3 [0.646 , -4.646] zefry@sas.edu.my Quadratic Equations 2.2 Forming a quadratic equation from given roots A. If the roots of a quadratic equation are known, such as then, the quadratic equation is (x – p)(x – q) = 0 x 2 – px – qx + pq = 0 x 2 – (p + q)x + pq = 0 x=p and x=q Notice that p + q = sum of roots ( SOR ) and pq = product of roots ( POR ) Hence, the quadratic equation with two given roots can be obtained as follows :x 2 – (SOR)x + (POR) = 0 Examples Form the quadratic equations from the given roots. 1. x = 1 , x = 2 Method 1 (x – 1)(x – 2) = 0 x2 - 2x – x + 2 = 0 x2 - 3x + 2 = 0 Method 2 SOR = 1 + 2 = 3 POR = 1 x 2 = 2 x2 – 3x +2 = 0 2. x = - 2 , x = 3 Exercise 4 Form the quadratic equations with the given roots. 1. x = 3 , x = 2 1 2. x = - 6 , 3 3. x = - 4 , x = - 6 4 4. x = -3 , x = 5 5. x = -7 , 3 [x2 - 5x + 6 = 0] 6. x = 5 only [x2 - 10x + 25 = 0] zefry@sas.edu.my [3x2 +17x - 6 = 0 ] [x2 + 10x + 24 = 0] [5x2 + 11x - 12=0 ] [x2 + 4x - 21 = 0] Quadratic Equations 1 3 1 1 8. x , x 2 3 [3x2 - x = 0] 7. x = 0 , x = [6x2 - 5x + 1 = 0] B. To find the S.O.R and P.O.R from the quadratic equation in general form ax 2 + bx + c = 0 b c a , x 2 + x = 0 a a Compare with Then , x 2 – (SOR)x + (POR) = 0 b a c POR = a SOR = If and are the roots of the quadratic equation ax2 + bx + c = 0, b then + = a c = a Activity 3 1. The roots for each of the following quadratic equations are and . Find the value of + and for the following equation Quadratic Equations a. x 2 – 12x + 4 = 0 b. x 2 = 4x + 8 zefry@sas.edu.my 12 4 Quadratic Equations c. 3 – 2x 2 = 10x d. 3x 2 + 8x = 10 e. 2x 2 + 3x + 4 = 0 C. Solving problems involving SOR and POR Activity 4 1. Given that and are the roots of the quadratic equation 2x 2 + 3x + 4 = 0 . Form a quadratic equation with roots 2 and 2. 2x 2 + 3x + 4 = 0 3 x2 x 2 0 2 3 = 2 =2 zefry@sas.edu.my New roots 3 SOR = 2 2 = 2 ( ) = 2 = -3 2 POR = 2 ( 2 ) 4 = 4(2) = 8 x 2 – (SOR)x + (POR) = 0 x 2 – (-3)x + 8 = 0 x 2 + 3x + 8 = 0 Quadratic Equations 2. If and are the roots of the quadratic equation 2x 2 – 5x – 1 = 0 , form a quadratic equation with roots 3 and 3. 3. Given that and are the roots of the quadratic equation 2x 2 – 3x + 4 = 0 . Form a 1 1 quadratic equation with roots and . 4. Given that m and n are roots of the quadratic equation 2x2 – 3x – 5 = 0 , form a quadratic equation which has the roots 2m n zefry@sas.edu.my and 2n m . Quadratic Equations Exercise 5 1. If and are roots of the quadratic equation 2x2 + 3x + 1 = 0, form a quadratic equation for the following roots a. 2 and 2 [x2 + 3x + 2 = 0] b. c. 2 d. 2. [x2 - 3x + 2 = 0 ] 2 + 3 and 2 + 3 and [8x2 + 6x + 1 = 0 ] 2 [x2 - 6x - 5 = 0] 2 - 1 and 2 - 1 If and are the roots of equation 2x equation with roots 2 and 2 . 2 – 5x – 6 = 0 , form a quadratic [ 4 x 2 5x 3 0 ] 3. Given that and are the roots of the equation 3x 2 = 4 – 9x , form a quadratic equation with roots 2 and 2 . [ 9 x 2 105x 16 0 ] 4. Given m and n are the roots of the equation x 2 + 10x – 2 = 0 , form a quadratic equation with roots; (a) 2m + 1 and 2n + 1 [ x 2 18x 27 0 ] 3 3 (b) and [ 2 x 2 30 x 9 0 ] m n 5. Given that and 3 are the roots of the equation x 2 + 2bx + 3a = 0 , prove that 4a = b 2 . 6. Given one of the root of the quadratic equation x 2 – 5kx + k = 0 is four times the other root, find the value of k . [k 1 ] 4 7. One of the roots of the quadratic equation 2x2 + 6x = 2k – 1 is twice the value of the other root whereby k is a constant. Find the roots and the value of k. [-1, -2 ; k = 3 ] 2 zefry@sas.edu.my Quadratic Equations 3. DISCRIMINANT OF A QUADRATIC EQUATIONS 3.1 Determining the types of roots of quadratic equations For the quadratic equation ax2 + bx + c = 0 , the value of b2 – 4ac will determine the types of roots. b2 – 4ac is called the “discriminant” Condition b – 4ac > 0 b2 – 4ac = 0 b2 – 4ac < 0 2 Type of roots Two different roots Two equal roots No roots Example Determine the type of roots for each of the following quadratic equations . (a) 2x2 – 7x + 9 = 0 (b) 2x2 – 3x – 9 = 0 a = 2 , b = -7 , c = 9 b2 – 4ac = (-7)2 – 4(2)(9) = 49 – 72 = -23 <0 no roots Exercise 6 Calculate the discriminant for each of the following quadratic equation and then state the type of roots for each equation . 1. x2 – 8x + 14 = 0 5. x(3x – 5) = 2x- 5 2. 2x2 – 7x – 1 = 0 6. 5(5 – 4x) = 4x2 zefry@sas.edu.my Quadratic Equations 3. 4 + x2 = 4x 7. x2 = 2 – 4x 4. (x – 2)2 = 3 8. 2x2 + 3x = 0 3.2 Solving problems involving the use of the discriminant Activity 5 1. The quadratic equation 2kx2 + 4x – 3 = 0 has two equal roots , find the value of k . 2. The quadratic equation x2 + 2kx + (k + 1)2 = 0 has real roots , find the range of values of k. 3. Show that the equation x2 + m + 1 = 8x has two different roots if m < 15 . zefry@sas.edu.my Quadratic Equations 4. The straight line y = tx – 2 is a tangent to the graph of a curve y = 2x2 + 4x , find the value of t (t > 0) . 5. Given that the quadratic equation p(x2 + 9) = - 5qx has two equal roots , find the ratio of p : q . Hence, solve those quadratic equation . 6. Show that the quadratic equation x2 + kx = 9 – 3k has real roots for all the value of k . zefry@sas.edu.my Quadratic Equations Exercise 7 1. Find the possible values of m if the quadratic equation (4 – 2m)x2 – 2m = 1 – 3mx has two equal roots . k 2. The equation x2 – 2x + = 0 has two different roots , find the range of values of k . 3 3. Given that the equation (p + 1)x2 – 2x + 5 = 0 has no roots , find the range of values of p . 4. Find the range value of k if the quadratic equation x2 + 1 = k – 4x has real roots . 5. The quadratic equation 2x(x – 3) = k – 2x has two distinct roots. Find the range of values of k. 6. The quadratic equation (m – 2)x 2 + 2x + 3 = 0 has two distinct roots. Find the range of values of m. 7. A quadratic equation 4x(x + 1) = 4x – 5mx – 1 has two equal roots. Find the possible values of m. 8. The straight line y = 2x – 1 does not intersect the curve y = 2x 2 + 3x + p. Find the range of values of p. 9. The straight line y = 6x + m does not intersect the curve y = 5 + 4x – x 2 . Find the range of values of m. 10. The straight line y = 2x + c intersect the curve y = x2 – x + 1 at two different points, find the range of values of c. 11. Find the range values of m if the straight line y = mx + 1 does not meet the curve y2 = 4x . 12. Show that the quadratic equation kx2 + 2(x + 1) = k has real roots for all the values of k. Answers for Exercise 7 1. m 4 , 4 7 4. k 3 7. m = 4 or m = 4 5 5 9. m > 6 k<3 3. p > 4 5. k > -2 6. m < 7 3 2. 5 8. p > – 3 8 10. . c > 5 4 zefry@sas.edu.my 11. m > 1 Quadratic Equations Enrichment Exercise – Quadratic Equations 1. The quadratic equation kx2 + 4x + 3= 0 has two different roots, find the range of values of k . 2. Find the possible values of k if the quadratic equation x2 + (2 + k)x + 2(2 + k) = 0 has two equal roots. 3. Show that the quadratic equation x2 + (2k – 1)x + k2 = 0 has real roots if k 1 . 4 4. Find the possible values of k if the straight line y = 2x + k is a tangent to the curve y = x2 + x + 1 . 5. Given that and are the roots of the quadratic equation 2x2 – 8x + 1 = 0 . Form the quadratic equation with roots 2 and 2 . 6. Solve each of the following quadratic equation :a. 6x2 + 5x – 4 = 0 b. y(y + 1) = 10 c. 2x(x + 5) = 7x + 2 d. 16x2 + 8x + 1 = 0 7. The roots of the equation 2ax2 + x + 3b = 0 are 3 4 and . Find the value of a and b. 2 3 8. If and are the roots of quadratic equation 2x2 – 3x – 6 = 0 , form the quadratic equation with roots and . 3 3 1 9. Given and – 5 are the roots of the quadratic equation . Write the quadratic equation 2 in the form of ax2 + bx + c = 0 . 10. Given that m + 2 and n – 1 are the roots of the equation x2 + 5x = - 4 . Find the possible values of m and n . 11. Given that 2 and m are the roots of the equation (2x – 1)(x + 3) = k(x – 1) such that k is a constant . Find the value of m and k . 12. Given one of the root of the equation 2x2 + 6x = 2k – 1 is twice the other root, such that k is a constant . Find the value of the roots and the value of k . 13. One of the root of the quadratic equation h + 2x – x2 = 0 is - 1 . Find the value of h. zefry@sas.edu.my Quadratic Equations 1 . Give your answer in the 2 form ax2 + bx + c = 0 , where a , b and c are constants. (SPM 2004) 14. Form the quadratic equation which has the roots -3 and 15. Solve the quadratic equation x(2x - 5) = 2x – 1 . Give your answer correct to three decimal places .(SPM 2005) 16. The straight line y = 5x – 1 does not intersect the curve y = 2x2 + x + p . Find the range of the values of p .(SPM 2005) 17. A quadratic equation x2 + px + 9 = 2x has two equal roots. Find the possible values of p.(SPM 2006) Answers on Enrichment Exercises 4 3 3 4. k = 4 1. k < 6. (a) 4 1 x , 3 2 7. a = 3 , b = -4 9. 2x 2 9x 5 0 2. k = 6 , - 2 5. (b) y = 2.702 , - 3.702 8. (c) 2 x 2 24 x 65 0 x 1 ,2 2 (d) 6 x 2 3x 2 0 10. n = 0 , - 3 ; m = - 6 , - 3 11. m = 3 , k = 15 12. roots = - 1 , -2 and k = 13. h = 3 14. 2x2 + 5x – 3 = 0 15. x = 3.35 , 0.15 16. p > 1 17. p = -8 , 4 zefry@sas.edu.my x 3 2 1 4