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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