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------------------------------------------------ Solve the equation using the quadratic formula:

[tex]15x^2 + 13x = 0[/tex]

Choose the best answer from the options provided:

A. [tex]x = -\frac{13}{15}, 0[/tex]
B. [tex]x = 0[/tex]
C. [tex]x = \frac{13}{15}, 0[/tex]
D. [tex]x = \pm \frac{13}{15}[/tex]

To solve the equation [tex]$15x^2 + 13x = 0$[/tex] using the quadratic formula, let's first write the equation in the standard form [tex]$ax^2 + bx + c = 0$[/tex]. For this equation:

- [tex]$a = 15$[/tex]
- [tex]$b = 13$[/tex]
- [tex]$c = 0$[/tex]

The quadratic formula is:

[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]

1. Calculate the Discriminant:
The discriminant is given by [tex]$b^2 - 4ac$[/tex].
[tex]\[
b^2 = 13^2 = 169
\][/tex]
[tex]\[
4ac = 4 \times 15 \times 0 = 0
\][/tex]
[tex]\[
\text{Discriminant} = 169 - 0 = 169
\][/tex]

2. Calculate the Roots:
Since the discriminant is non-negative, we can proceed with finding the roots.

- Root 1:
[tex]\[
x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} = \frac{-13 + \sqrt{169}}{30} = \frac{-13 + 13}{30} = \frac{0}{30} = 0
\][/tex]

- Root 2:
[tex]\[
x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a} = \frac{-13 - \sqrt{169}}{30} = \frac{-13 - 13}{30} = \frac{-26}{30} = -\frac{13}{15}
\][/tex]

Therefore, the solutions to the equation are [tex]$x = 0$[/tex] and [tex]$x = -\frac{13}{15}$[/tex].

The best answer from the choices provided is:
A. [tex]$x = -\frac{13}{15}, 0$[/tex]