To tackle this problem, we start by working with the given system of equations:
1. [tex]\(a + b + c = 13\)[/tex]
2. [tex]\(ab + bc + ca = 84\)[/tex]
Our goal is to find the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]. This problem can be approached systematically by solving these simultaneous equations.
We'll use some algebraic manipulation and the properties of polynomial roots to derive the solution.
Consider a polynomial whose roots are [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]. A polynomial with these roots can be written as:
[tex]\[ P(x) = (x - a)(x - b)(x - c) \][/tex]
Expanding this polynomial, we get:
[tex]\[ P(x) = x^3 - (a+b+c)x^2 + (ab + bc + ca)x - abc \][/tex]
Given the equations:
[tex]\[ a + b + c = 13 \][/tex]
[tex]\[ ab + bc + ca = 84 \][/tex]
Substituting these known values into the polynomial, we have:
[tex]\[ P(x) = x^3 - 13x^2 + 84x - abc \][/tex]
From this representation, we need to solve for [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] directly using these values. Utilizing algebraic equations can sometimes be complex, especially if they involve square roots or quadratic terms which can't be simplified easily by hand.
Next, let's consider the solutions that were derived from solving the system:
[tex]\[ \left( -\frac{c}{2} - \frac{\sqrt{-3c^2 + 26c - 167}}{2} + \frac{13}{2}, -\frac{c}{2} + \frac{\sqrt{-3c^2 + 26c - 167}}{2} + \frac{13}{2}, c \right) \][/tex]
[tex]\[ \left( -\frac{c}{2} + \frac{\sqrt{-3c^2 + 26c - 167}}{2} + \frac{13}{2}, -\frac{c}{2} - \frac{\sqrt{-3c^2 + 26c - 167}}{2} + \frac{13}{2}, c \right) \][/tex]
Each tuple consists of [tex]\((a, b, c)\)[/tex] which are valid solutions to the given equations.
Therefore, if we consider one of the solutions:
[tex]\[ \left( -\frac{c}{2} - \frac{\sqrt{-3c^2 + 26c - 167}}{2} + \frac{13}{2}, -\frac{c}{2} + \frac{\sqrt{-3c^2 + 26c - 167}}{2} + \frac{13}{2}, c \right) \][/tex]
And the other solution:
[tex]\[ \left( -\frac{c}{2} + \frac{\sqrt{-3c^2 + 26c - 167}}{2} + \frac{13}{2}, -\frac{c}{2} - \frac{\sqrt{-3c^2 + 26c - 167}}{2} + \frac{13}{2}, c \right) \][/tex]
Both of these tuples uphold the initial conditions given in the problem, making them the correct values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] that satisfy the equations.
So, when [tex]\((a + b + c = 13)\)[/tex] and [tex]\((ab + bc + ca = 84)\)[/tex], the specific values for variables [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are given by these solutions involving [tex]\(c\)[/tex] in a complex quadratic form.
In summary:
[tex]\[ a, b, c = \left( -\frac{c}{2} - \frac{\sqrt{-3c^2 + 26c - 167}}{2} + \frac{13}{2}, -\frac{c}{2} + \frac{\sqrt{-3c^2 + 26c - 167}}{2} + \frac{13}{2}, c \right) \][/tex]
or
[tex]\[ a, b, c = \left( -\frac{c}{2} + \frac{\sqrt{-3c^2 + 26c - 167}}{2} + \frac{13}{2}, -\frac{c}{2} - \frac{\sqrt{-3c^2 + 26c - 167}}{2} + \frac{13}{2}, c \right) \][/tex]
