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2 CHAPTER : QUADRATIC EQUATIONS 1.1 Recognize Quadratic Equations and express it in general form Activity 1 General form ax bx + c = 0, where a, b and c are constants, a 0 Properties 1. Equation must be in one unknown only. The highest power of the unknown is Examples 1. x + 3x 1 = 0 is a quadratic equation. x 9 = 0 is a quadratic equation 3. 8x 3 x = 0 is not a quadratic equation 1. Determine whether each of the following equation is a quadratic equation or not. Equations Answer (a) x x = 0 Yes (b) x y = 0 (c) 3x + = 0 (d) m 7m 3 = 0 (e) k k = 0 (f) y = 0. Rewrite each of the following quadratic equation in the general form. State the value of a, b and c. Quadratic equations (a) 1 + x = x(x + 3) 1 + x = x + 3x x + x 1 = 0 Value of a, b and c a = 1 b = 1 c = -1 (b) m = 1 m

3 (c) (y + 6)(y ) = - 7 (d) x = 7 x 3 (e) (x + 1) = Roots of Quadratic Equations Notes 1. The root of a quadratic equation is the value(number) of the unknown(variable) that satisfy the equation.. 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. (a) x x = 0 ; x = - 1, 1, (b) x + 7x + 3 = 0 ; x = - 3, - 1, 1, 3. Determine by inspection which of the values of x are roots of the following quadratic equations. (a) (x + 3)(x ) = 0 ; x = 3,, - 3 (b) x(x + ) = 0 ; x =, 0, - 3. If x = is the root of the quadratic equation x 3kx -10 = 0, find the value of k.

4 . SOLVING QUADRATIC EQUATION.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. If p = q the equation have two equal roots (one root only) 3. The equation must be written in general form ax + bx + c = 0 before factorization. Activity Solve the following quadratic equations by factorization. 1. x 7x 8 = 0. x x + = 0 ( x 8 ) ( x + 1 ) = 0 x 8 = 0 or x + 1 = 0 x = 8 or x = x 8x = 0. x 9 = x + 13x 5 = 0 6. (3x + 1)(x - 1) = x x 5 x 8. (x + 1)(x 5) = 16 x x 5 = 16 x x 1 = 0 ( x 7 ) ( x 3 ) = 0 x = 7 or x = 3

5 t t t 10. (p + 1)(p + 1) = 0 Exercise 1 Solve the following quadratic equation by factorisation. 1. x 5x 6 = 0 [6,-1] 9. x 9x + 0 = 0 [5,]. m + 5m = 0 [-8,3] 10. x 13x + 3 = 0 [, 3 1 ] 1 7, ] 3 3. y + 10y + = 0 [-6,-] 11. x 3 = 5x [, 3. x + 3x 5 = 0 [1, ] 1. 6x 11x = 7 [ x 6x 7 = 0 [ 7 ] 13. (x 3) = 9 [ 5,-], 8 6. a + a = 0 [0.-] 1. (3m + 1)(m 1) = 7 [ n = 0 [ ] x + = 13x [, 3 1 3, ] 3 1 ], 5 8. (x + 1)(x + 3 ) = 0 [, 3] 16. x(x + ) = 1 [ -7,3] B. By Completing the Square Notes 1. The expression x x + 1 can be written in the form (x 1) This is called perfect square. Example Solve each of the following quadratic equation (a) (x + 1) = 9 (b) x = 9 x + 1 = 3 x + 1 = 3, x + 1 = -3 x =, x = - (c) (x + ) = 36

6 . 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 x + hx into a perfect square, we add the term h ( ) to the expression. And this will make x hx x h hx h x. To solve the equation by using completing the square method for quadratic equation ax + hx + k = 0, follow this steps ; Step 1 : Rewrite the equation in the form ax + hx = - k Step : If the coefficient of x is 1, reduce the coefficient to 1 (by dividing). Step 3 : Add ( h ) to both sides of the equation. Step : Write the expression on the LHS as perfect square. Step 5 : Solve the equation Examples 1. x + 6x 9 = 0. x 5x 8 = 0 x + 6x = 9 x x + = 9 + ( x + 3 ) = 18 x + 3 = 18 x + 3 =.3 x =.3 3, x = x = 1.3, x = -7.3 Exercise Solve the following equations by completing the square. (Give your answers correct to four significant figures) 1. x 8x + 1 = 0 [5.1,.59]. x 7x 1 = 0 [3.6, -0.1] 3. x + 5x + 1 = 0 [-0.09,-.79]. x 3x + 5 = 0 [-.19,1.19] 5. x = 5(x + ) [7.6, -.6] 6. -x 1x + 3 = 0 [-3.3,0.3] 7. x 3x = 0 [.35,-0.85]

7 C. By Using the quadratic formula The quadratic equation ax + bx + c can be solved by using the quadratic formula x = b b ac a, where a 0 Example x 7x 3 = 0 a =, b = -7, c = -3 ( 7) x x ( 7) () ()( 3) x = 3.886, Exercise 3 Use the quadratic formula to find the solutions of the following equations. Give your answers correct to three decimal places. 1. x 3x 5 = 0 [.193, ]. 9x = x 16 [1.333 ] 3. x + 5x 1 = 0 [0.186, -.686]. 3x + 1x 9 = 0 [.899, ] x x = 0 [0.768, -0.3] 6. m = 0 m [0.573, -5.39] 7. k 1 k 3 [-1.10, 6.10] 8. x(x + ) = 3 [0.66, -.66]

8 . Forming a quadratic equation from given roots A. If the roots of a quadratic equation are known, such as x = p and x = q then, the quadratic equation is (x p)(x q) = 0 x px qx + pq = 0 x (p + q)x + pq = 0 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 (SOR)x + (POR) = 0 Examples Form the quadratic equations from the given roots. 1. x = 1, x = Method 1 Method (x 1)(x ) = 0 SOR = 1 + = 3 x - x x + = 0 POR = 1 x = x - 3x + = 0 x 3x + = 0. x = -, x = 3 Exercise Form the quadratic equations with the given roots. 1. x = 3, x = [x - 5x + 6 = 0] 1. x = - 6, 3 [3x +17x - 6 = 0 ] 3. x = -, x = - 6 [x + 10x + = 0]. x = -3, x = 5 [5x + 11x - 1=0 ] 5. x = -7, 3 [x + x - 1 = 0] 6. x = 5 only [x - 10x + 5 = 0]

9 7. x = 0, x = 1 3 [3x - x = 0] x, x [6x - 5x + 1 = 0] 3 B. To find the S.O.R and P.O.R from the quadratic equation in general form ax + bx + c = 0 a, x + b c x = 0 a a Compare with x (SOR)x + (POR) = 0 Then, SOR = POR = c a b a If and are the roots of the quadratic equation ax + 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 1x + = 0 1 b. x = x + 8

10 c. 3 x = 10x d. 3x + 8x = 10 e. x + 3x + = 0 C. Solving problems involving SOR and POR Activity 1. Given that and are the roots of the quadratic equation x + 3x + = 0. Form a quadratic equation with roots and. x + 3x + = 0 New roots 3 3 x x 0 SOR = = ( ) = = -3 3 = POR = ( ) = () = 8 = x (SOR)x + (POR) = 0 x (-3)x + 8 = 0 x + 3x + 8 = 0

11 . If and are the roots of the quadratic equation x 5x 1 = 0, form a quadratic equation with roots 3 and Given that and are the roots of the quadratic equation x 3x + = 0. Form a quadratic equation with roots 1 and 1.. Given that m and n are roots of the quadratic equation x 3x 5 = 0, form a quadratic equation which has the roots m and n. n m

12 Exercise 5 1. If and are roots of the quadratic equation x + 3x + 1 = 0, form a quadratic equation for the following roots a. and [x + 3x + = 0] b. + 3 and + 3 [x - 3x + = 0 ] c. and d. - 1 and - 1 [x - 6x - 5 = 0] [8x + 6x + 1 = 0 ]. If and are the roots of equation x 5x 6 = 0, form a quadratic equation with roots and. [ x 5 x 3 0 ] 3. Given that and are the roots of the equation 3x = 9x, form a quadratic equation with roots and. [ 9 105x 16 0 x ]. Given m and n are the roots of the equation x + 10x = 0, form a quadratic equation with roots; (a) m + 1 and n + 1 [ x 18x 7 0] (b) 3 m and 3 n [ x 30x 9 0 ] 5. Given that and 3 are the roots of the equation x + bx + 3a = 0, prove that a = b. 6. Given one of the root of the quadratic equation x 5kx + k = 0 is four times the other root, find the value of k. [ 1 k ] 7. One of the roots of the quadratic equation x + 6x = k 1 is twice the value of the other root whereby k is a constant. Find the roots and the value of k. [-1, - ; k = 3 ]

13 3. DISCRIMINANT OF A QUADRATIC EQUATIONS 3.1 Determining the types of roots of quadratic equations For the quadratic equation ax + bx + c = 0, the value of b ac will determine the types of roots. b ac is called the discriminant Condition b ac > 0 b ac = 0 b ac < 0 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) x 7x + 9 = 0 (b) x 3x 9 = 0 Exercise 6 a =, b = -7, c = 9 b ac = (-7) ()(9) = 9 7 = -3 < 0 no roots Calculate the discriminant for each of the following quadratic equation and then state the type of roots for each equation. 1. x 8x + 1 = 0 5. x(3x 5) = x- 5. x 7x 1 = (5 x) = x

14 3. + x = x 7. x = x. (x ) = 3 8. x + 3x = 0 3. Solving problems involving the use of the discriminant Activity 5 1. The quadratic equation kx + x 3 = 0 has two equal roots, find the value of k.. The quadratic equation x + kx + (k + 1) = 0 has real roots, find the range of values of k. 3. Show that the equation x + m + 1 = 8x has two different roots if m < 15.

15 . The straight line y = tx is a tangent to the graph of a curve y = x + x, find the value of t (t > 0). 5. Given that the quadratic equation p(x + 9) = - 5qx has two equal roots, find the ratio of p : q. Hence, solve those quadratic equation. 6. Show that the quadratic equation x + kx = 9 3k has real roots for all the value of k.

16 Exercise 7 1. Find the possible values of m if the quadratic equation ( m)x m = 1 3mx has two equal roots.. The equation x x + 3 k = 0 has two different roots, find the range of values of k. 3. Given that the equation (p + 1)x x + 5 = 0 has no roots, find the range of values of p.. Find the range value of k if the quadratic equation x + 1 = k x has real roots. 5. The quadratic equation x(x 3) = k x has two distinct roots. Find the range of values of k. 6. The quadratic equation (m )x + x + 3 = 0 has two distinct roots. Find the range of values of m. 7. A quadratic equation x(x + 1) = x 5mx 1 has two equal roots. Find the possible values of m. 8. The straight line y = x 1 does not intersect the curve y = x + 3x + p. Find the range of values of p. 9. The straight line y = 6x + m does not intersect the curve y = 5 + x x. Find the range of values of m. 10. The straight line y = x + c intersect the curve y = x 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 y = x. 1. Show that the quadratic equation kx + (x + 1) = k has real roots for all the values of k. Answers for Exercise 7 1. m,. k < 3 3. p > 7 5. k 3 5. k > - 6. m < m = or m = 5 8. p > m > c > m > 1

17 Enrichment Exercise Quadratic Equations 1. The quadratic equation kx + x + 3= 0 has two different roots, find the range of values of k.. Find the possible values of k if the quadratic equation x + ( + k)x + ( + k) = 0 has two equal roots. 3. Show that the quadratic equation x + (k 1)x + k = 0 has real roots if k 1.. Find the possible values of k if the straight line y = x + k is a tangent to the curve y = x + x Given that and are the roots of the quadratic equation x 8x + 1 = 0. Form the quadratic equation with roots and. 6. Solve each of the following quadratic equation :- a. 6x + 5x = 0 b. y(y + 1) = 10 c. x(x + 5) = 7x + d. 16x + 8x + 1 = 0 7. The roots of the equation ax + x + 3b = 0 are 3 and. Find the value of a and b If and are the roots of quadratic equation x 3x 6 = 0, form the quadratic equation with roots 3 and Given 1 and 5 are the roots of the quadratic equation. Write the quadratic equation in the form of ax + bx + c = Given that m + and n 1 are the roots of the equation x + 5x = -. Find the possible values of m and n. 11. Given that and m are the roots of the equation (x 1)(x + 3) = k(x 1) such that k is a constant. Find the value of m and k. 1. Given one of the root of the equation x + 6x = k 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 + x x = 0 is - 1. Find the value of h.

18 1. Form the quadratic equation which has the roots -3 and 1. Give your answer in the form ax + bx + c = 0, where a, b and c are constants. (SPM 00) 15. Solve the quadratic equation x(x - 5) = x 1. Give your answer correct to three decimal places.(spm 005) 16. The straight line y = 5x 1 does not intersect the curve y = x + x + p. Find the range of the values of p.(spm 005) 17. A quadratic equation x + px + 9 = x has two equal roots. Find the possible values of p.(spm 006) Answers on Enrichment Exercises 1. k < 3. k = 3. k = 6, - 5. x x (a) 1 x, (b) y =.70, (c) 1 1 x, (d) x 3 7. a = 3, b = x 3x 0 9. x 9x n = 0, - 3 ; m = - 6, m = 3, k = roots = - 1, - and k = 13. h = 3 1. x + 5x 3 = x = 3.35, p > p = -8, 3

SOLUTION OF QUADRATIC EQUATIONS LESSON PLAN. A3 Topic Overview ALGEBRA

ALGEBRA A Topic Overview A SOLUTION OF QUADRATIC EQUATIONS This topic describes three methods of solving Quadratic equations. assumes you understand and have practised using the algebraic methods described