One of the roots of the equation x ^ 2 + tx + 27 = 0 is (-9). Find the second root and coefficient t.

Let us determine the coefficient t by substituting the well-known root x1 = – 9 into the equation:

(- 9) ^ 2 – 9t + 27 = 0,

81 – 9t +27 = 0,

9t = 108,

t = 12.

Substitute the obtained value of the coefficient t into the equation and find the roots of the equation. In this case, one of the roots should turn out to be x1 = – 9.

x ^ 2 + 12x + 27 = 0.

Let’s define the discriminant:

D = 12 ^ 2 – 4 * 1 * 27 = 144 – 108 = 36,

the root of the resulting discriminant is 6,

x1,2 = (- 12 ± 6) / 2,

x1 = (- 12 – 6) / 2 = – 9,

x2 = (- 12 + 6) / 2 = -3.

Answer: t = 12, x2 = -3.