If alpha and beta zeroes of 3x²-x-4 find the value of alpha⁴beta³ + alpha³beta⁴.[Ans: -64/81]​

EXPLANATION.

α,β are the zeroes of the Quadratic Equation,

⇒ p(x) = 3x² - x - 4.

As we know that,

Sum of zeroes of the quadratic equation,

⇒ α + β = -b/a.

⇒ α + β = -(-1)/3 = 1/3.

Products of zeroes of the quadratic equation,

⇒ αβ = c/a.

⇒ αβ = -4/3.

To find value of = (α⁴β³ + α³β⁴).

⇒ (α⁴β³ + α³β⁴).

⇒ α³β³(α + β).

⇒ (αβ)³(α + β).

⇒ (-4/3)³(1/3).

⇒ (-64/27).(1/3).

⇒ (-64/81).

Value of (α⁴β³ + α³β⁴) = -64/81.

                                                                                                                                       

MORE INFORMATION.Location of Roots of a quadratic Equation ax² + bx + c = 0.

(A) = Conditions for both the roots will be greater than k.

(1) = D ≥ 0.

(2) = k < -b/2a.

(3) = af(k) > 0.

(B) = Conditions for both the roots will be less than k.

(1) = D ≥ 0.

(2) = k > -b/2a.

(3) = af(k) > 0.

(C) = Conditions for k lie between the roots.

(1) = D > 0.

(2) = af(k) < 0.

(D) = Conditions for exactly one roots lie in the interval (k₁, k₂) where k₁ < k₂.

(1) = f(k₁)f(k₂) < 0.

(2) = D > 0.

(E) = When both roots lie in the interval (k₁, k₂) where k₁ < k₂.

(1) = D > 0.

(2) = f(k₁).f(k₂) > 0.

(F) = Any algebraic expression f(x) = 0 in interval [a, b] if,

(1) = sign of f(a) and f(b) are of same than either no roots or even no. of roots exists.

(2) = sign of f(a) and f(b) are opposite then f(x) = 0 has at least one real roots or odd no. of roots.