Question

the equation whose roots are the cube of the roots of the equation ax²+bx+c=0​

Answer 1

Let α and β be the roots of the given equation ax² + bx + c = 0

We know that

Sum of roots of the equation = α + β = - Coefficient of x / Coefficient of x² = - b/a

⇒ α + β = - b/a

Product of roots of the equation = αβ = Constant / Coefficient of x² = c/a

⇒ αβ = c/a    

If the cubes of the roots are roots of an equation

α³, β³ will be the roots of the equation

Sum of roots of the equation = α³ + β³  

Since x³ + y³ = ( x + y )³ - 3xy( x + y )

⇒  α³ + β³ = ( α + β )³ - 3αβ( α + β )  

⇒ α³ + β³ = ( - b/a )³ - 3( c/a )( - b/a )

⇒ α³ + β³ = - b³/a³ + 3bc/a²

⇒ α³ + β³ = ( - b³ + 3abc) / a³  

Product of roots of the equation = α³β³ = ( αβ )³ = ( c/a )³ = c³/a³

Quadratic equation :

x² - ( Sum of roots )x + Product of roots = 0

Hence,  the equation whose roots are the cube of the roots of the equation ax² + bx + c = 0

$\Rightarrow \rm x^2 - \Bigg(\dfrac{- b^3 + 3abc }{a^3} \Bigg)x+ \dfrac{c^3}{a&^3} = 0$

Multiplying every term with a³

⇒ a³x² -  ( - b³ + 3abc )x + c³ = 0

⇒ a³x² + ( b³ - 3abc )x + c³ = 0

∴ the equation whose roots are the cube of the roots of the equation ax² + bx + c is  a³x² + ( b³ - 3abc )x + c³ = 0

Answer 2

AnswEr:-

Let α and β be the roots of the given equation ax² + bx + c = 0

Concept:-

→Sum of roots of the equation = α + β = - Coefficient of x / Coefficient of x² = - b/a

⇒ α + β = - b/a

→Product of roots of the equation = αβ = Constant / Coefficient of x² = c/a

⇒ αβ = c/a    

→If the cubes of the roots are roots of an equation

α³, β³ will be the roots of the equation

→Sum of roots of the equation = α³ + β³  

Since x³ + y³ = ( x + y )³ - 3xy( x + y )

⇒  α³ + β³ = ( α + β )³ - 3αβ( α + β )  

⇒ α³ + β³ = ( - b/a )³ - 3( c/a )( - b/a )

⇒ α³ + β³ = - b³/a³ + 3bc/a²

⇒ α³ + β³ = ( - b³ + 3abc) / a³  

→Product of roots of the equation = α³β³ = ( αβ )³ = ( c/a )³ = c³/a³

Quadratic equation :

x² - ( Sum of roots )x + Product of roots = 0

→Hence,  the equation whose roots are the cube of the roots of the equation ax² + bx + c = 0

⇒ a³x² -  ( - b³ + 3abc )x + c³ = 0

⇒ a³x² + ( b³ - 3abc )x + c³ = 0

∴ the equation whose roots are the cube of the roots of the equation ax² + bx + c is  a³x² + ( b³ - 3abc )x + c³ = 0

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