write quadratic equations when sum of zeroes and product of zeroes are -5,3
Answers
Required Quadratic equation:-
x² +5x + 3
Given:-
- Sum of zeros of the Quadratic equation is -5
- Product of zeros of the Quadratic equation is 3
To find :-
Quadratic equation
Solution:-
As we know that ,
If α, β are the roots of Quadratic equation then the required Quadratic equation is
● x² - (α+β) x + αβ
● α+ β = Sum of zeros
● αβ = Product of zeros
According to the Question,
α+ β = -5
αβ = 3
Substituting in formula,
x² - (α+β) x + αβ
x² -(-5) x + 3
x² + 5(x) + 3
x² + 5x + 3
So, the required Quadratic equation is x² + 5x + 3
Verification:-
As we got the Quadratic equation Hence Sum of zeros must be -5 and product of zeros must be 3
Firstly lets find zeros of the Quadratic equation
By Quadratic formula
x² + 5x + 3 = 0
x = -b± √(b²-4ac) /2a
Comparing with general form of Quadratic equation in order to get values of a, b ,c
- a = 1
- b = 5
- c =3
x = -5 ± √(5)² -4(1)(3) /2(1)
x = -5± √25-12 /2
x = -5±√13 /2
x =( -5+√13)/2 , (-5-√13)/2 ,
So, zeros are (-5+√13)/2 , (-5-√13)/2
Sum of zeros = -5
(-5+√13)/2 + (-5-√13)/2 = -5
(-5 -5) /2 = -5
(-10)/2 = -5
-5 = -5 (Verified)
Product of zeros = 3
(-5+√13/2 ) (-5-√13/2) = 3
Applying (a + b)(a-b) = a²-b²
(-5/2 +√13/2) (-5/2 -√13/2) =3
(-5/2)² - (√13/2)² = 3
25/4 - 13/4 = 3
25-13/4 = 3
12/4 = 3