Question

If a, b, and c are solutions to the

equation 3x3 – 25x2 - 50x = 0 and
a < b < c, evaluate 10c – 6a.

Answer 1

`\displaystyle 10c - 6a = 110`

Explanation:

We are given the equation:

`\displaystyle 3x^3 - 25x^2 -50x = 0`

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:

`\displaystyle \begin{aligned} 3x^3 - 25x^2 -50x & = 0 \\ \\ x(3x^2 - 25x -50) & = 0 \\ \\ x(3x+5)(x-10) & = 0 \end{aligned}`

From the Zero Product Property:

`\displaystyle x = 0 \text{ or } 3x + 5 = 0 \text{ or } x - 10 = 0`

Solve for each case:

`\displaystyle x = 0 \text{ or } x = -(5)/(3) \text{ or } x = 10`

We can see that -5/3 < 0 < 10. Thus, a = -5/3, b = 0, and c = 10.

Therefore:

`\displaystyle \begin{aligned} 10c - 6a & = 10(10) - 6\left(-(5)/(3)\right) \\ \\ &= 100 + 10 \\ \\ & = 110 \end{aligned}`