Question

3. If α , β are the roots of the equation 3x2 + 7 x − 2 = 0 , find the values of (i) (α/β) + (β/α) (ii) (α2/β) + (β2/α)

Answer 1

Solution

Given :-

• α & β are roots of equation 3x² + 7x - 2 = 0 ________(1)

Find :-

• Value of (α/β) + (β/α)
• Value of (α2/β) + (β2/α)

Explanation

Some Formula

★ Sum of roots = -(Coefficient of x)/(Coefficient of x²)

★ Product of roots = (Constant part)/(Coefficient of x²)

So, Now

➡ Sum of roots = -7/3

➡ α + β = -7/3__________(2)

Again,

➡ Product of roots = -2/3

➡α . β = -2/3____________(3)

Now, Squaring both side of equ(2)

➡ (α + β)² = (-7/3)²

➡α² + β² + 2α.β = 49/9

Keep Value by equ(3)

➡α² + β² = 49/9 - 2 × (-2/3)

➡α² + β² = 49/9 + 4/3

➡α² + β² = (49 + 12)/9

➡α² + β² = 61/9___________(4)

Now, First Calculate

➡ (α/β) + (β/α)

➡ (α² + β²)/αβ

Keep value by equ(3) &(4)

➡ (61/9)/(-2/3)

➡61/9 × -3/2

➡ -61/6. ( Ans.)

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Now, Calculate Second.

➡(α²/β) + (β²/α)

➡(α³ + β³)/β . α

[ (α³ + β³) = (α+ β)(α²+ β² - β . α)

➡[(α+ β)(α²+ β² - β . α)]/β . α

Keep values by equ(2) , (3) &(4)

➡[(-7/3){61/9 - (-2/3)}]/(-2/3)

➡(-7/3) × (61+6)/9 × -3/2

➡7 × 67/18

➡469/18. [ Ans.]

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