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

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

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]

Please select the best answer from the choices provided:
A
B
C
D

Sure! Let's solve the quadratic equation [tex]$15x^2 + 13x = 0$[/tex] using the quadratic formula.

The quadratic formula is given by:

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

For the equation [tex]$15x^2 + 13x = 0$[/tex], the coefficients are:
- [tex]$a = 15$[/tex]
- [tex]$b = 13$[/tex]
- [tex]$c = 0$[/tex]

Step 1: Calculate the Discriminant

The discriminant [tex]$(b^2 - 4ac)$[/tex] is essential to use the quadratic formula:

[tex]\[
b^2 - 4ac = 13^2 - 4 \times 15 \times 0
\][/tex]
[tex]\[
b^2 - 4ac = 169 - 0 = 169
\][/tex]

Step 2: Use the Quadratic Formula

We plug the values into the quadratic formula:

[tex]\[
x = \frac{-13 \pm \sqrt{169}}{2 \times 15}
\][/tex]

Since [tex]$\sqrt{169} = 13$[/tex], we continue:

[tex]\[
x = \frac{-13 \pm 13}{30}
\][/tex]

Step 3: Calculate the Two Solutions

1. For [tex]$x_1$[/tex]:

[tex]\[
x_1 = \frac{-13 + 13}{30} = \frac{0}{30} = 0
\][/tex]

2. For [tex]$x_2$[/tex]:

[tex]\[
x_2 = \frac{-13 - 13}{30} = \frac{-26}{30} = -\frac{13}{15}
\][/tex]

Conclusion:

The solutions to the equation [tex]$15x^2 + 13x = 0$[/tex] are [tex]$x = 0$[/tex] and [tex]$x = -\frac{13}{15}$[/tex].

Thus, the best choice from the given options is A: [tex]$x = -\frac{13}{15}, 0$[/tex].