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As we know that  Let the distance x and time taken be t.  Now According to the question, And From (1) we have Putting this on (2), we get Since speed can never be negative, we choose, Hence the speed of the train is 40 km/hour.  
L.H.S.  ,  and R.H.S  can be written as: i.e.,  This equation is of type: . Hence, the given equation is a quadratic equation.
Let us assume the length and breadth of the park be  respectively. Then, the perimeter will be  The area of the park is: Given :  Comparing to get the values of a, b and c. The value of the discriminant  As . Therefore, this equation will have two equal roots. And hence the roots will be: Therefore, the length of the park,  and breadth of the park .  
Let the age of one friend be  and the age of another friend will be:  4 years ago, their ages were,  and . According to the question, the product of their ages in years was 48.   or   Now, comparing to get the values of . Discriminant value  As . Therefore, there are no real roots possible for this given equation and hence, This situation is NOT possible.    
Let the breadth of mango grove be . Then the length of mango grove will be . And the area will be: Which will be equal to  according to question. Comparing to get the values of . Finding the discriminant value: Here,  Therefore, the equation will have real roots. And hence finding the dimensions: As negative value is not possible, hence the value of breadth of mango grove will be...
For two equal roots for the quadratic equation:  The value of the discriminant . Given equation:  Can be written as:  Comparing and getting the values of a,b, and, c. The value of  But  is NOT possible because it will not satisfy the given equation. Hence the only value of  is 6 to get two equal roots.  
For two equal roots for the quadratic equation:  The value of discriminant . Given equation:  Comparing and getting the values of a,b, and, c. The value of  Or,   
The value of the discriminant  The discriminant > 0. Therefore the given quadratic equation has two distinct real root roots are So the roots are  
Here the value of discriminant =0, which implies that roots exist and the roots are equal. The roots are given by the formula So the roots are
For a quadratic equation,   the value of discriminant determines the nature of roots and is equal to: If D>0 then roots are distinct and real. If D<0 then no real roots. If D= 0 then there exists two equal real roots. Given the quadratic equation,  . Comparing with general to get the values of a,b,c. Finding the discriminant: Here D is negative hence there are no real roots possible for the...
Let the sides of the squares be .              (NOTE: length are in meters) And the perimeters will be:  respectively. Areas  respectively. It is given that,                     .................................(1)                        .................................(2) Solving both equations:   or    putting in equation (1), we obtain Solving by the factorizing method: Here the roots...
Let the average speed of the passenger train be . Given the average speed of the express train  also given that the time taken by the express train to cover 132 km is 1 hour less than the passenger train to cover the same distance. Therefore,  Can be written as quadratic form: Roots are:  As the speed cannot be negative. Therefore, the speed of the passenger train will be  and  The speed...
Let the time taken by the smaller pipe to fill the tank be  Then, the time taken by the larger pipe will be: . The fraction of the tank filled by a smaller pipe in 1 hour:   The fraction of the tank filled by the larger pipe in 1 hour. Given that two water taps together can fill a tank in  hours. Therefore, Making it a quadratic equation: Hence the roots are  As the time taken cannot...
Let the speed of the train be  Then, time taken to cover  will be:  According to the question, Making it a quadratic equation. Now, solving by the factorizing method: However, the speed cannot be negative hence, The speed of the train is .
Given the difference of squares of two numbers is 180. Let the larger number be 'x' and the smaller number be 'y'. Then, according to the question:   and   On solving these two equations: Solving by the factorizing method: As the negative value of x is not satisfied in the equation:  Hence, the larger number will be 18 and a smaller number can be found by,  putting x = 18, we...
Let the shorter side of the rectangle be x m. Then, the larger side of the rectangle wil be . Diagonal of the rectangle: It is given that the diagonal of the rectangle is 60m more than the shorter side. Therefore,  Solving by the factorizing method: Hence, the roots are:  But the side cannot be negative. Hence the length of the shorter side will be: 90 m  and the length of the larger...
Let the marks obtained in Mathematics be 'm' then, the marks obtain in English will be '30-m'. Then according to the question: Simplifying to get the quadratic equation: Solving by the factorizing method: We have two situations when, The marks obtained in Mathematics is 12, then marks in English will be 30-12 = 18.   Or, The marks obtained in Mathematics is 13, then marks in English will...
Let the present age of Rehman be  years. Then, 3 years ago, his age was  years. and 5 years later, his age will be  years. Then according to the question we have, SImplifying it to get the quadratic equation: Hence the roots are:  However, age cannot be negative Therefore, Rehman is 7 years old in present.  
Given equation:  So, simplifying it,     or     Can be written as: Hence the roots of the given equation are:
Given equation:  So, simplifying it, Comparing with the general form of the quadratic equation: , we get Now, applying the quadratic formula to find the roots: Therefore, the roots are        
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