the sum of two roots of quadratic equation is 5 and sum of their cude is 35 find the equation
Answers
Answer:
x² - 5x + 6 = 0
Step-by-step explanation:
Given---> Sum of two roots of quadratic equation is 5 and sum of their cube is 35.
To find---> Quadratic equation
Solution---> Let roots of quadratic equation be α and β .
ATQ, Sum of roots of quadratic equation is 5
=> α + β = 5
ATQ, Sum of cubes of roots = 35
=> α³ + β³ = 35
=> ( α )³ + ( β )³ = 35
We have an identity , as follows,
( a³ + b³ ) = ( a + b ) ( a² + b² - ab ) , applying it , we get,
=> ( α + β ) ( α² + β² - αβ ) = 35
Adding and subtracting 2αβ in second bracket.
=> ( 5 ) { ( α² + β² + 2αβ ) - 2αβ - αβ } = 35
=> ( α + β )² - 3αβ = 35 / 5
=> ( 5 )² - 3αβ = 7
=> 25 - 3αβ = 7
=> - 3αβ = 7 - 25
=> - 3αβ = - 18
=> 3αβ = 18
=> αβ = 18 / 3
=> αβ = 6
We know that quadratic equation can be represented as
x² - ( Sum of roots ) x + product of roots = 0
=> x² - ( α + β ) x + ( αβ ) = 0
=> x² - ( 5 ) x + 6 = 0
=> x² - 5x + 6 = 0
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