10th Class Mathematics Quadratic Equations Relations between Roots of Quadratic Equation

      Relations between the Roots of the Quadratic Equation

If  are the roots of the quadratic equation, then the relation between the roots of the quadratic equation is given by,             

Sum of the roots

Product of the roots  

           Formation of Quadratic Equations

If  are the roots of the quadratic equation, S denotes its sum and P denotes its product, then the quadratic equation is given by:

           Graphical Representation of a Quadratic Equation

For the quadratic equation , the nature of graph for different values of D is: (a) If D < 0, and a > 0, then the graph is given by:  

If a < 0, then the graph is given by.  

(b) If D = 0, and a > 0, then the graph of the function is given by  

If a < 0, then the graph is given by  

(d) If D > 0, and a > 0, then the graph of the function is given by,  

If a < 0, then the graph of the function is given by,               

           Roots of Biquadratic Equation

Any biquadratic equation,  , will have four roots. If a, P, Y, and 5 are its roots, then the relation between the roots is given by Sum of the roots

Sum of product of two roots at a time

Sum of product of three roots at a time

Product of the roots  

  Find the value of.

(a)                                          

(b)

(c)                                               

(d)

(e) None of these  

Answer: (a)  

Explanation  

Let      

  A If  are the roots of the equation, then the value of  are.

(a)                                                        

(b)  

(c)                                                    

(d)

(e) None of these  

Answer: (b)

Explanation

We have,  

  If  is real and  is the roots of , then find the value of .

(a)                                        

(b)

(c)                                           

(d)

(e) None of these  

Answer: (d)  

Explanation 

We have sum of the roots

Product of the roots

Form the above two equations we get,

  If  are the roots of and  are the roots of, then find the value of .

(a) 15                                                       

(b) 33

(c) 41                                                       

(d) 63

(e) None of these  

Answer: (b)  

  If the coefficient of z in the quadratic equation  is taken as 18 in place of 12 and its roots were found to be -16 and -2. The roots of the original equation are:

(a) - 12 & - 6                                          

(b) - 14 & - 4

(c) - 16 & - 2                                          

(d) ? 8 &- 4

(e) None of these  

Answer: (d)          

• The demerit of the factorization method, is that it takes time to figure out the numbers within the factors.
• The first known solution of a quadratic equation is the one given in the Berlin papyrus from the Middle Kingdom (ca. 2160-1700 BC) in Egypt.
• The Hindu mathematician Aryabhata (475 or 476-550) gave a rule for the sum of a geometric series that shows knowledge of the quadratic equations with both solutions
• Viete was among the first to replace geometric methods of solution with analytic ones, although he apparently did not grasp the idea of a general quadratic equation.
• Sridhara gave the positive root of the quadratic formula, as stated by Bhaskara.      

• The most general quadratic equation is in the form, where a, b, c are constants and x is the variable.
• A real number m is said to be the root of the quadratic equation if it satisfies the quadratic equation, .
• For any quadratic equation, the number of root is always two.
• Quadratic equation can be solved by factorization or using the discriminant method.
• The value of .
• No real root exist for D < 0.