FIND QUADRATIC EQUATION WHEN ROOTS ARE GIVEN

If the sum and product of the roots of a quadratic equation is given, we can construct the quadratic equation as shown below.  

x2 - (sum of roots) x + product of roots  =  0

(or)

x2 - (a+β)x + aβ  =  0

Form a quadratic equation whose roots are

(i)  3, 4

(ii) 3+√7, 3-√7

(iii) (4+√7)/2 , (4-√7)/2

Question 1 :

3, 4

Solution :

α = 3, β = 4

x2-(α+β)x+αβ  =  0

Sum of roots :

α+β  =  3+4 ==> 7

Product of roots :

αβ  =  3(4) ==> 12

By applying those values in the general form we get,

x2-7x+12  =  0

Question 2 :

3 + √7 , 3 - √7

Solution :

α  =  3+√7, β = 3 - √7

Sum of roots :

α+β  =  3+√7+3-√7 

  =  6

Product of roots :

α β  =  (3+√7)(3-√7)

=  32 - 7

=  9 - 7

= 2

By applying those values in the general form we get,

x2-6x+2  =  0

Question 3 :

(4+√7)/2 , (4-√7)/2

Solution :

α = (4 + √7)/2, β = (4 - √7)/2

x2-(α+β)x+αβ  =  0

Sum of roots :

α + β  =  (4 + √7)/2 + (4 - √7)/2

=  (4 + √7 + 4 - √7)/2

=  8/2

  =  4

Product of roots :

α β = [(4 + √7)/2] [(4 - √7)/2] 

= (42 - (√7)2)/4

= (16 - 7)/4

= 9/4

by applying those values in the general form we get,

x2-4x+(9/4)  =  0

4x2-16x+9  =  0

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