Answer:
[tex]\displaystyle 10c - 6a = 110[/tex]
Step-by-step explanation:
We are given the equation:
[tex]\displaystyle 3x^3 - 25x^2 -50x = 0[/tex]
Where a, b, and c are solutions to the equation and where a < b < c, we want to determine the value of 10c - 6a.
To find the solutions of the equation, we can factor:
[tex]\displaystyle \begin{aligned} 3x^3 - 25x^2 -50x & = 0 \\ \\ x(3x^2 - 25x -50) & = 0 \\ \\ x(3x+5)(x-10) & = 0 \end{aligned}[/tex]
From the Zero Product Property:
[tex]\displaystyle x = 0 \text{ or } 3x + 5 = 0 \text{ or } x - 10 = 0[/tex]
Solve for each case:
[tex]\displaystyle x = 0 \text{ or } x = -\frac{5}{3} \text{ or } x = 10[/tex]
We can see that -5/3 < 0 < 10. Thus, a = -5/3, b = 0, and c = 10.
Therefore:
[tex]\displaystyle \begin{aligned} 10c - 6a & = 10(10) - 6\left(-\frac{5}{3}\right) \\ \\ &= 100 + 10 \\ \\ & = 110 \end{aligned}[/tex]
