Question

if alpha be a root of equation 4x^2+2x-1=0 then prove that 4alpha^3-3alpha is other root

Answer 1

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Answer 2

To Prove: 4alpha^3-3alpha is other root

Given: equation is 4x^2+2x−1=0

Step-by-step explanation:

Suppose that α and β are the equation's roots.

Let’s suppose one root is α and another root is β= 4 α ^3-3 α

We know

In a quadratic equation, the sum of roots is given as

α+β= -½            

β=-½-α                      ……..(1)

We are given quadratic equation i.e.4x^2+2x−1=0

As α is one the root, the equation becomes

4 α ^2+2 α −1=0

4 α ^2= 1- 2α          ………….(2)

We need to prove β= 4 α ^3-3 α

4α^2.α- 3a

Inserting (2) in above equation

(1- 2α)α - 3α  

α - 2α^2-3α  

-2α  ^2- 2α  

Multiplying and dividing by 2

½(-4α  ^2- 4α  )

Inserting (2) in the above equation

-½(1- 2α + 4α)

-½(1+2α)

-½-α

From the equation (1) β=-½-α

Hence, proved that 4alpha^3-3alpha is other root

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