Question

If the roots of x^4 - 10x^3 + 37x^2 - 60x + 36 = 0 are alpha, alpha, beta, beta (alpha < beta), then 2 alpha + 3 beta - 2alpha beta is ___
(ans - 1)
(need explanation)
*EAMCET 2018*

Answer 1

Answer :-

We have:-

x⁴ - 10x³ + 37x² - 60x + 36 = 0

Since, α, α and β, are the roots of the above equation,

⟹ α + α + β + β = -(10)/1 = 10 - (1)

Sum of zeroes = -b/a

⟹ (α)(α)(β)(β) = 36/1 = 36 - (2)

Product of zeroes = e/a

From (1) and (2), we have:-

⟹ α + β = 5 - (1)

• β = 5 - α - (3)

⟹ α²β² = 36

• α² - 5α + 6 = 0 - (4)

⟹ αβ = 6 - (2)

From (3) and (2), again we have:-

⟹ α(5 - α) = 6 ⟹ α² - 5α + 6 = 0

⟹ α = 2, 3 ⟹ β = 3, 2

Since, α < β, hence α = 2, β = 3.

∴ 2α + 3β - 2αβ will be:-

⟹ 2(2) + 3(3) - 2(2)(3)

⟹ 4 + 9 - 12

⟹ 1

∴The value of α and β is 2, 3 and the value of 2α + 3β - 2αβ is 1.

Answer 2

EXPLANATION.

Roots : x⁴ - 10x³ + 37x² - 60x + 36 = 0.

⇒ α, α, β, β (α < β).

As we know that,

Sum of the zeroes of the biquadratic polynomial.

⇒ α + α + β + β = - b/a.

⇒ α + α + β + β = -(-10)/1 = 10.

⇒ 2α + 2β = 10.

⇒ α + β = 5. - - - - - (1).

⇒ α = 5 - β. - - - - - (1).

Products of the zeroes of the biquadratic polynomial.

⇒ (α)(α)(β)(β) = e/a.

⇒ α x α x β x β = 36.

⇒ α²β² = 36.

⇒ (αβ)² = (6)².

⇒ αβ = 6.

Put the values of α = 5 - β in the equation, we get.

⇒ (5 - β)(β) = 6.

⇒ 5β - β² = 6.

⇒ 6 + β² - 5β = 0.

⇒ β² - 5β + 6 = 0.

Factorizes the equation into middle term splits, we get.

⇒ β² - 3β - 2β + 6 = 0.

⇒ β(β - 3) - 2(β - 3) = 0.

⇒ (β - 2)(β - 3) = 0.

⇒ β = 2   and   β = 3.

Put the values of β in the equation (1), we get.

⇒ α = 5 - β.

Put the value of β = 2 in the equation, we get.

⇒ α = 5 - 2.

⇒ α = 3.

Put the value of β = 3 in the equation, we get.

⇒ α = 5 - 3.

⇒ α = 2.

⇒ α = 2  and  β = 3.

⇒ α = 3  and  β = 2.

It is clearly given that,

⇒ (α < β).

So, we take.

⇒ α = 2  and  β = 3.

To find :

⇒ 2α + 3β - 2αβ.

Put the values in the equation, we get.

⇒ 2(2) + 3(3) - 2(2)(3).

⇒ 4 + 9 - 12.

⇒ 13 - 12 = 1.

⇒ 2α + 3β - 2αβ = 1.

                                                                                                                   

MORE INFORMATION.

(1) = If D₁ + D₂ ≥ 0 : At least one of the equation has real roots.

(2) = If D₁ + D₂ < 0 : At least one of the equation has imaginary roots.

(3) = If D₁. D₂ < 0 : One equation has real and distinct roots and other has imaginary roots.