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3 7 MATHEMATICS Therefore, their product = (x 5) (40 x) =40x x x = x + 45x 00 So, x + 45x 00 = 4 (Given that product = 4) i.e., x + 45x 34 = 0 i.e., x 45x + 34 = 0 Therefore, the number of marbles John had, satisfies the quadratic equation x 45x + 34 = 0 which is the required representation of the problem mathematically. (ii) Let the number of toys produced on that day be x. Therefore, the cost of production (in rupees) of each toy that day = 55 x So, the total cost of production (in rupees) that day = x (55 x) Therefore, x (55 x) = 750 i.e., 55x x = 750 i.e., x + 55x 750 = 0 i.e., x 55x = 0 Therefore, the number of toys produced that day satisfies the quadratic equation x 55x = 0 which is the required representation of the problem mathematically. Example : Check whether the following are quadratic equations: (i) (x ) + = x 3 (ii) x(x + ) + 8 = (x + ) (x ) (iii) x (x + 3) = x + (iv) (x + ) 3 = x 3 4 Solution : (i) LHS = (x ) + = x 4x = x 4x + 5 Therefore, (x ) + = x 3 can be rewritten as x 4x + 5 = x 3 i.e., x 6x + 8 = 0 It is of the form ax + bx + c = 0. Therefore, the given equation is a quadratic equation (--04)

6 QUADRATIC EQUATIONS 75 Note that we have found the roots of x 5x + 3 = 0 by factorising x 5x + 3 into two linear factors and equating each factor to zero. Example 4 : Find the roots of the quadratic equation 6x x = 0. Solution : We have 6x x = 6x + 3x 4x =3x (x + ) (x + ) =(3x )(x + ) The roots of 6x x = 0 are the values of x for which (3x )(x + ) = 0 Therefore, 3x = 0 or x + = 0, i.e., x = 3 or x = Therefore, the roots of 6x x = 0 are and. 3 We verify the roots, by checking that and satisfy 6x x = 0. 3 Example 5 : Find the roots of the quadratic equation 3x 6x 0. Solution : 3x 6x = 3x 6x 6x = 3x 3x 3x = 3x 3x So, the roots of the equation are the values of x for which Now, 3x 0 for x x x. 3 So, this root is repeated twice, one for each repeated factor 3x. Therefore, the roots of 3x 6x 0 are 3, (--04)

7 76 MATHEMATICS Example 6 : Find the dimensions of the prayer hall discussed in Section 4.. Solution : In Section 4., we found that if the breadth of the hall is x m, then x satisfies the equation x + x 300 = 0. Applying the factorisation method, we write this equation as x 4x + 5x 300 = 0 x (x ) + 5 (x ) = 0 i.e., (x )(x + 5) = 0 So, the roots of the given equation are x = or x =.5. Since x is the breadth of the hall, it cannot be negative. Thus, the breadth of the hall is m. Its length = x + = 5 m. EXERCISE 4.. Find the roots of the following quadratic equations by factorisation: (i) x 3x 0 = 0 (ii) x + x 6 = 0 (iii) x 7x 5 0 (iv) x x + 8 = 0 (v) 00 x 0x + = 0. Solve the problems given in Example. 3. Find two numbers whose sum is 7 and product is Find two consecutive positive integers, sum of whose squares is The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 3 cm, find the other two sides. 6. A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was ` 90, find the number of articles produced and the cost of each article. 4.4 Solution of a Quadratic Equation by Completing the Square In the previous section, you have learnt one method of obtaining the roots of a quadratic equation. In this section, we shall study another method. Consider the following situation: The product of Sunita s age (in years) two years ago and her age four years from now is one more than twice her present age. What is her present age? To answer this, let her present age (in years) be x. Then the product of her ages two years ago and four years from now is (x )(x + 4) (--04)

8 QUADRATIC EQUATIONS 77 Therefore, (x )(x + 4) = x + i.e., x + x 8 = x + i.e., x 9 = 0 So, Sunita s present age satisfies the quadratic equation x 9 = 0. We can write this as x = 9. Taking square roots, we get x = 3 or x = 3. Since the age is a positive number, x = 3. So, Sunita s present age is 3 years. Now consider the quadratic equation (x + ) 9 = 0. To solve it, we can write it as (x + ) = 9. Taking square roots, we get x + = 3 or x + = 3. Therefore, x = or x = 5 So, the roots of the equation (x + ) 9 = 0 are and 5. In both the examples above, the term containing x is completely inside a square, and we found the roots easily by taking the square roots. But, what happens if we are asked to solve the equation x + 4x 5 = 0? We would probably apply factorisation to do so, unless we realise (somehow!) that x + 4x 5 = (x + ) 9. So, solving x + 4x 5 = 0 is equivalent to solving (x + ) 9 = 0, which we have seen is very quick to do. In fact, we can convert any quadratic equation to the form (x + a) b = 0 and then we can easily find its roots. Let us see if this is possible. Look at Fig. 4.. In this figure, we can see how x + 4x is being converted to (x + ) 4. Fig (--04)

9 78 MATHEMATICS The process is as follows: x + 4x =(x + 4 x ) + 4 x =x + x + x = (x + ) x + x =(x + ) x + x + =(x + ) x + (x + ) =(x + ) (x + ) =(x + ) 4 So, x + 4x 5 = (x + ) 4 5 = (x + ) 9 So, x + 4x 5 = 0 can be written as (x + ) 9 = 0 by this process of completing the square. This is known as the method of completing the square. In brief, this can be shown as follows: So, x + 4x = x x 4 x + 4x 5 = 0 can be rewritten as 4 x 45 =0 i.e., (x + ) 9 = 0 Consider now the equation 3x 5x + = 0. Note that the coefficient of x is not a perfect square. So, we multiply the equation throughout by 3 to get 9x 5x + 6 = 0 Now, 9x 5x + 6 = = 5 (3 x) 3x (3 x) 3x 6 = 5 5 3x 6 4 = 5 3x (--04)

10 QUADRATIC EQUATIONS 79 So, 9x 5x + 6 = 0 can be written as i.e., 5 3x 4 5 3x =0 = 4 So, the solutions of 9x 5x + 6 = 0 are the same as those of 5 3x. 4 i.e., 3x 5 = or 5 3x = (We can also write this as 5 3x, where ± denotes plus minus.) Thus, 3x = 5 or So, x = 5 or x 5 x 6 6 Therefore, x = or x = 4 6 i.e., x = or x = 3 Therefore, the roots of the given equation are and. 3 Remark : Another way of showing this process is as follows : The equation 3x 5x + = 0 is the same as 5 x x =0 3 3 Now, x 5 x = x (--04)

11 80 MATHEMATICS = = 5 5 x x x So, the solutions of 3x 5x + = 0 are the same as those of 5 x 0, 6 6 which are x 5 6 = ± 6, i.e., x = 5 = and x = 5 = Let us consider some examples to illustrate the above process. Example 7 : Solve the equation given in Example 3 by the method of completing the square. Solution : The equation x 5 3 5x + 3 = 0 is the same as x x 0. Now, 5 3 x x = x = x Therefore, x 5x + 3 = 0 can be written as x So, the roots of the equation x 5x + 3 = 0 are exactly the same as those of 5 5 x 0. Now, x =0 is the same as Therefore, 5 x = 4 4 i.e., x = i.e., x = 5 or x i.e., x = 3 or x = 5 x (--04)

12 QUADRATIC EQUATIONS 8 Therefore, the solutions of the equations are Let us verify our solutions. 3 x and. 3 Putting x 3 3 in x 5x + 3 = 0, we get 5 3 0, which is correct. Similarly, you can verify that x = also satisfies the given equation. In Example 7, we divided the equation x 5x + 3 = 0 throughout by to get x 5 x 3 = 0 to make the first term a perfect square and then completed the square. Instead, we can multiply throughout by to make the first term as 4x = (x) and then complete the square. This method is illustrated in the next example. Example 8 : Find the roots of the equation 5x 6x = 0 by the method of completing the square. Solution : Multiplying the equation throughout by 5, we get 5x 30x 0 = 0 This is the same as (5x) (5x) = 0 i.e., (5x 3) 9 0 = 0 i.e., (5x 3) 9 = 0 i.e., (5x 3) =9 i.e., 5x 3 = 9 i.e., 5x = 3 9 So, x = Therefore, the roots are Verify that the roots are and and (--04)

13 8 MATHEMATICS Example 9 : Find the roots of 4x + 3x + 5 = 0 by the method of completing the square. Solution : Note that 4x + 3x + 5 = 0 is the same as (x) + (x) =0 i.e., 3 9 x =0 i.e., x =0 i.e., x 3 4 = But x cannot be negative for any real value of x (Why?). So, there is 4 no real value of x satisfying the given equation. Therefore, the given equation has no real roots. Now, you have seen several examples of the use of the method of completing the square. So, let us give this method in general. Consider the quadratic equation ax + bx + c = 0 (a 0). Dividing throughout by b c a, we get x x 0 a a This is the same as b b c x 0 a a a i.e., b b 4ac x = 0 a 4a So, the roots of the given equation are the same as those of b b 4ac x 0, a 4a i.e., those of b b 4ac x a 4a () 05-6 (--04)

14 QUADRATIC EQUATIONS 83 If b 4ac 0, then by taking the square roots in (), we get b x = a b 4ac a Therefore, x = b b 4ac a b b 4ac b b 4ac So, the roots of ax + bx + c = 0 are and, if a a b 4ac 0. If b 4ac < 0, the equation will have no real roots. (Why?) Thus, if b 4ac 0, then the roots of the quadratic equation ax + bx + c = 0 are given by b± b 4ac a This formula for finding the roots of a quadratic equation is known as the quadratic formula. Let us consider some examples for illustrating the use of the quadratic formula. Example 0 : Solve Q. (i) of Exercise 4. by using the quadratic formula. Solution : Let the breadth of the plot be x metres. Then the length is (x + ) metres. Then we are given that x(x + ) = 58, i.e., x + x 58 = 0. This is of the form ax + bx + c = 0, where a =, b =, c = 58. So, the quadratic formula gives us the solution as x = i.e., x = 4()(58) or x i.e., x = 6 or x = Since x cannot be negative, being a dimension, the breadth of the plot is 6 metres and hence, the length of the plot is 33m. You should verify that these values satisfy the conditions of the problem (--04)

15 84 MATHEMATICS Example : Find two consecutive odd positive integers, sum of whose squares is 90. Solution : Let the smaller of the two consecutive odd positive integers be x. Then, the second integer will be x +. According to the question, x + (x + ) = 90 i.e., x + x + 4x + 4 = 90 i.e., x + 4x 86 = 0 i.e., x + x 43 = 0 which is a quadratic equation in x. Using the quadratic formula, we get x = i.e., x = or x = 3 But x is given to be an odd positive integer. Therefore, x 3, x =. Thus, the two consecutive odd integers are and 3. Check : + 3 = + 69 = 90. Example : A rectangular park is to be designed whose breadth is 3 m less than its length. Its area is to be 4 square metres more than the area of a park that has already been made in the shape of an isosceles triangle with its base as the breadth of the rectangular park and of altitude m (see Fig. 4.3). Find its length and breadth. Solution : Let the breadth of the rectangular park be x m. So, its length = (x + 3) m. Therefore, the area of the rectangular park = x(x + 3) m = (x + 3x) m. Now, base of the isosceles triangle = x m. Therefore, its area = x = 6 x m. According to our requirements, x + 3x =6x + 4 i.e., x 3x 4 = 0 Using the quadratic formula, we get Fig (--04)

16 QUADRATIC EQUATIONS 85 x = 3 5 But x (Why?). Therefore, x = 4. = 3 5 = 4 or So, the breadth of the park = 4m and its length will be 7m. Verification : Area of rectangular park = 8 m, area of triangular park = 4 m = (8 4) m Example 3 : Find the roots of the following quadratic equations, if they exist, using the quadratic formula: (i) 3x 5x + = 0 (ii) x + 4x + 5 = 0 (iii) x x + = 0 Solution : (i) 3x 5x + = 0. Here, a = 3, b = 5, c =. So, b 4ac = 5 4 = > 0. Therefore, x = 5 5, i.e., x = or x = So, the roots are 3 and. (ii) x + 4x + 5 = 0. Here, a =, b = 4, c = 5. So, b 4ac = 6 0 = 4 < 0. Since the square of a real number cannot be negative, therefore b 4ac will not have any real value. So, there are no real roots for the given equation. (iii) x x + = 0. Here, a =, b =, c =. So, b 4ac = 8 8 = 0 Therefore, x = 0 0, 4 i.e., x. So, the roots are, (--04)

17 86 MATHEMATICS Example 4 : Find the roots of the following equations: (i) Solution : x 3, x 0 (ii) 3, x 0, x x x (i) x 3. Multiplying throughout by x, we get x x + = 3x i.e., x 3x + = 0, which is a quadratic equation. Here, a =, b = 3, c = So, b 4ac = 9 4 = 5 > 0 Therefore, x = 3 5 (Why?) So, the roots are 3 5 and 3 5. (ii) 3, x 0,. x x As x 0,, multiplying the equation by x (x ), we get (x ) x =3x (x ) =3x 6x So, the given equation reduces to 3x 6x + = 0, which is a quadratic equation. Here, a = 3, b = 6, c =. So, b 4ac = 36 4 = > 0 Therefore, x = So, the roots are 3 3 and (--04)

18 QUADRATIC EQUATIONS 87 Example 5 : A motor boat whose speed is 8 km/h in still water takes hour more to go 4 km upstream than to return downstream to the same spot. Find the speed of the stream. Solution : Let the speed of the stream be x km/h. Therefore, the speed of the boat upstream = (8 x) km/h and the speed of the boat downstream = (8 + x) km/h. The time taken to go upstream = distance 4 speed 8 x hours. 4 Similarly, the time taken to go downstream = 8 x According to the question, hours x 8 x = i.e., 4(8 + x) 4(8 x) = (8 x) (8 + x) i.e., x + 48x 34 = 0 Using the quadratic formula, we get x = = = 4860 = 6 or Since x is the speed of the stream, it cannot be negative. So, we ignore the root x = 54. Therefore, x = 6 gives the speed of the stream as 6 km/h. EXERCISE 4.3. Find the roots of the following quadratic equations, if they exist, by the method of completing the square: (i) x 7x + 3 = 0 (ii) x + x 4 = 0 (iii) 4x 4 3x 3 0 (iv) x + x + 4 = 0. Find the roots of the quadratic equations given in Q. above by applying the quadratic formula (--04)

19 88 MATHEMATICS 3. Find the roots of the following equations: (i) x 3, x 0 (ii), x 4, 7 x x4 x The sum of the reciprocals of Rehman s ages, (in years) 3 years ago and 5 years from now is. Find his present age In a class test, the sum of Shefali s marks in Mathematics and English is 30. Had she got marks more in Mathematics and 3 marks less in English, the product of their marks would have been 0. Find her marks in the two subjects. 6. The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field. 7. The difference of squares of two numbers is 80. The square of the smaller number is 8 times the larger number. Find the two numbers. 8. A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken hour less for the same journey. Find the speed of the train. 9. Two water taps together can fill a tank in 9 3 hours. The tap of larger diameter takes 0 8 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank. 0. An express train takes hour less than a passenger train to travel 3 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is km/h more than that of the passenger train, find the average speed of the two trains.. Sum of the areas of two squares is 468 m. If the difference of their perimeters is 4 m, find the sides of the two squares. 4.5 Nature of Roots In the previous section, you have seen that the roots of the equation ax + bx + c = 0 are given by x = b b 4ac a If b 4ac > 0, we get two distinct real roots b b 4ac. a a b b 4ac and a a 05-6 (--04)

20 QUADRATIC EQUATIONS 89 b b b If b 4ac = 0, then x = 0, i.e., x or a a a b So, the roots of the equation ax + bx + c = 0 are both a Therefore, we say that the quadratic equation ax + bx + c = 0 has two equal real roots in this case. If b 4ac < 0, then there is no real number whose square is b 4ac. Therefore, there are no real roots for the given quadratic equation in this case. Since b 4ac determines whether the quadratic equation ax + bx + c = 0 has real roots or not, b 4ac is called the discriminant of this quadratic equation. So, a quadratic equation ax + bx + c = 0 has (i) two distinct real roots, if b 4ac > 0, (ii) two equal real roots, if b 4ac = 0, (iii) no real roots, if b 4ac < 0. Let us consider some examples. Example 6 : Find the discriminant of the quadratic equation x 4x + 3 = 0, and hence find the nature of its roots. Solution : The given equation is of the form ax + bx + c = 0, where a =, b = 4 and c = 3. Therefore, the discriminant b 4ac = ( 4) (4 3) = 6 4 = 8 < 0 So, the given equation has no real roots. Example 7 : A pole has to be erected at a point on the boundary of a circular park of diameter 3 metres in such a way that the differences of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 metres. Is it possible to do so? If yes, at what distances from the two gates should the pole be erected? Solution : Let us first draw the diagram (see Fig. 4.4). Let P be the required location of the pole. Let the distance of the pole from the gate B be x m, i.e., BP = x m. Now the difference of the distances of the pole from the two gates = AP BP (or, BP AP) = 7 m. Therefore, AP = (x + 7) m. Fig (--04)

21 90 MATHEMATICS Now, AB = 3m, and since AB is a diameter, APB = 90 (Why?) Therefore, AP + PB =AB (By Pythagoras theorem) i.e., (x + 7) + x =3 i.e., x + 4x x = 69 i.e., x + 4x 0 = 0 So, the distance x of the pole from gate B satisfies the equation x + 7x 60 = 0 So, it would be possible to place the pole if this equation has real roots. To see if this is so or not, let us consider its discriminant. The discriminant is b 4ac = 7 4 ( 60) = 89 > 0. So, the given quadratic equation has two real roots, and it is possible to erect the pole on the boundary of the park. Solving the quadratic equation x + 7x 60 = 0, by the quadratic formula, we get x = = Therefore, x = 5 or. Since x is the distance between the pole and the gate B, it must be positive. Therefore, x = will have to be ignored. So, x = 5. Thus, the pole has to be erected on the boundary of the park at a distance of 5m from the gate B and m from the gate A. Example 8 : Find the discriminant of the equation 3x x + 3 the nature of its roots. Find them, if they are real. Solution : Here a = 3, b = and c. 3 = 0 and hence find Therefore, discriminant b 4ac = ( ) = 4 4 = 0. Hence, the given quadratic equation has two equal real roots. The roots are b, b,,, i.e.,, i.e.,. a a (--04)

23 9 MATHEMATICS A NOTE TO THE READER In case of word problems, the obtained solutions should always be verified with the conditions of the original problem and not in the equations formed (see Examples, 3, 9 of Chapter 3 and Examples 0,, of Chapter 4) (--04)

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