find the nature of roots of x²-3√5x+10=0 quadratic equation if the real root exist find them
Answers
Answer:
the nature of the root is irrational and un equal
Step-by-step explanation:
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Given : Quadratic equation is x²- 3√5x + 10 = 0
To find : Nature of roots of the equation
Solution :
In order to find the nature of roots of any quadratic equation, we use the concept of discriminant. Discriminant helps us to find the nature of roots of any quadratic equation.
Nature of roots can be determined by values of Discriminant as follows :-
- If discriminant is +ve, two real distinct roots exists for the equation.
- If discriminant is -ve, no real roots exists for the equation.
- If discriminant is 0, both the roots are real and are equal to each other.
General form of quadratic equation is:-
- ax² + bx + c = 0
Discriminant of any equation can be obtained by using following formula :-
- D = b² - 4ac
Here,
- D = Discriminant
- a = coefficient of x²
- b = coefficient of x
- c = Constant term
In the given equation,
- a = 1
- b = -3√5
- c = 10
Let's apply discriminant formula to find the nature of roots !
=> D = b² - 4ac
=> D = (-3√5)² - 4 (1) ( 10 )
=> D = 45 - 40
=> D = 5
Since D is positive, two real roots exists for the equation.
Let's find both of them !
=> x² - 3√5 x + 10 = 0
=> x² - 2√5 x - √5 x + 10 = 0
=> x ( x - 2 √5 ) - √5 ( x - 2 √5 ) = 0
=> ( x - 2 √5 ) ( x - √5 ) = 0
=> x = 2√5 , √5
So the roots of the given quadratic equations are 2√5 and √5.
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