Question

Find the quadratic equation whose roots are the reciprocals of the roots of
${x}^{2} + 4x - 10 = 0$




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Answer 1

Answer:

10 x² - 4x - 1 = 0

Step-by-step explanation:

Given---> x² + 4 x - 10 = 0

To find---> Quadratic equation whose roots are the reciprocal of the roots of given equation.

Solution---> Let α and β are roots of given equation

x² + 4x - 10 = 0

We know that,

Sum of roots = - Coefficient of x / Coefficient of x²

=> α + β = - 4 / 1

=> α + β = -4

Product of roots = Costant term / Coefficient of x²

=> α β = - 10 / 1

=> α β = - 10

Now, ATQ, roots of required quadratic equation is reciprocal of given quadratic equation.

So , roots of required equation is 1 / α and 1 / β .

Sum of roots = ( 1 / α ) + ( 1 / β )

= ( α + β ) / α β

= ( - 4 ) / ( - 10 )

= 4 / 10

Product of roots = ( 1 / α ) ( 1 / β )

= 1 / α β

= 1 / α β

= 1 / ( - 10 )

= - 1 / 10

We know that quadratic equation is,

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

=> x² - ( 4 / 10 ) x + ( - 1 / 10 ) = 0

=> 10 x² - 4x - 1 = 0