Answer:
13 years
Step-by-step explanation:
Let x years be the age of each twins this year, then x+5 years is the age of their older sister this year.
The product of the ages of three sibling is
[tex]x\cdot x\cdot (x+5)=x^3 +5x^2[/tex]
The sum of their ages is
[tex]x+x+x+5=3x+5[/tex]
Since the product of the three siblings' ages is exactly 2998 more than the sum of their ages, we have
[tex]x^3 +5x^2-(3x+5)=2,998\\ \\x^3 +5x^2-3x-5-2,998=0\\ \\x^3+5x^2-3x-3,003=0[/tex]
The divisors of 3,003 are
[tex]\pm 1, \pm 3,\pm 7, \pm 11,\pm 13, \pm 21, \pm 33,\pm 77,....[/tex] and so on.
Check positive (the age cannot be negative) numbers to be equation's solutions:
[tex]13^3+5\cdot 13^2-3\cdot 13-3,003=2,197+845-39-3,003=0[/tex]
So,
[tex]x^3+5x^2-3x-3,003=(x-13)(x^2+18x+231)=0[/tex]
The quadratic equation has no real solutions, because its discriminant
[tex]D=18^2-4\cdot 231=324-924=-600<0[/tex]
So, the twins are 13 years old (and the sister is 18 years old)
