Question

If a, ß are the root of the equation x2+px+q=0 find the value of a3 +B3?​

Answer 1

EXPLANATION.

α,β are the roots of the equation,

f(x) = x² + px + q = 0.

Sum of zeroes of quadratic equation.

⇒ α + β = -b/a.

⇒ α + β = -p.  .......(1).

Products of zeroes of quadratic equation.

⇒ αβ = c/a.

⇒ αβ = q. ........(2).

To find value of (α³ + β³).

Formula of (a³ + b³) = (a + b)(a² + b² - ab).

Similarly,

⇒ (α³ + β³) = (α + β)(α² + β² - αβ)

As we know that,

Formula of (α² + β²),

⇒ (α² + β²) = (α + β)² - 2αβ.

Put this value in equation, we get.

⇒ (α + β)[(α + β)² - 2αβ - αβ].

⇒ (α + β)[(α + β)² - 3αβ].

⇒ (-p)[(-p)² - 3(q)].

⇒ (-p)[p² - 3q].

⇒ -p³ + 3pq.

⇒ 3pq - p³.

Value of (α³ + β³) = 3pq - p³.