Find the value of k

Question

The two imaginary solutions of (x-1)(x-2)(x-3)=(6-1)(6-2)(6-3) satisfy the equation x^2 + k = 0. What is the value of k?

Answer 1

 (x-1)(x-2)(x-3)=(6-1)(6-2)(6-3)  simplify both sides by multiplying

(x² -3x +2)(x - 3) = (5)(4)(3)

x³ -6 x² + 11x - 6 = 60 subtract 60 from both sides

x³ - 6x² + 11x - 66 = 0 (if you have learned the sign change rules, f(-x) has no sign change, so there are

no neg. real solutions)

Use synthetic division and trial and error until you get

6| 1 - 6 11 -66

6 0 -66

1 0 11 0

So the quadratic factor is x² + 11 =0

Therefore k = 11