Question

X3 + bx2 + cx + d = 0 where b , c and d are real constants. The three roots of the cubic are denoted by α , β and γ . Given that α + β + γ = 4 and α2 + β2 + γ2 = 20 find the value of b

Answer 1

b = -4

Explanation:

A cubic equation is an equation in the form of ax³ + bx² + cx + d = 0, where a ≠ 0, and a, b, c, d are complex numbers.

If p, q and r are roots of the quadratic function, then:

p + q + r = -b/a

pq + pr + qr = c/a

pqr = -d/a

For a cubic equation x³ + bx² + cx + d with roots α , β and γ, then:

α + β + γ = -b/a

αβ + αγ + βγ = c/a

αβγ = -d/a

From the equation we can see that a = 1, hence:

α + β + γ = -b/1

4 = -b

b = -4

Also, α² + β² + γ² = (α + β + γ)² - 2(αβ + αγ + βγ)

Substituting:

20 = (4)² - 2(c/a)

20 = 16 - 2(c/1)

2c = 16 - 20

2c = -4

c = -2