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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]

Answer :

To solve the equation [tex]\(15x^2 + 13x = 0\)[/tex], we can use the quadratic formula. However, another straightforward approach is to factor the equation.

1. Factor the Equation:

The given equation is:
[tex]\[
15x^2 + 13x = 0
\][/tex]

We can factor out [tex]\(x\)[/tex] from both terms:
[tex]\[
x(15x + 13) = 0
\][/tex]

2. Apply the Zero Product Property:

According to the zero product property, if [tex]\(ab = 0\)[/tex], then either [tex]\(a = 0\)[/tex] or [tex]\(b = 0\)[/tex]. So, we set each factor equal to zero:

- [tex]\(x = 0\)[/tex]
- [tex]\(15x + 13 = 0\)[/tex]

3. Solve for [tex]\(x\)[/tex]:

- The first solution is straightforward: [tex]\(x = 0\)[/tex].

- For the second equation:
[tex]\[
15x + 13 = 0
\][/tex]
Subtract 13 from both sides:
[tex]\[
15x = -13
\][/tex]
Divide both sides by 15:
[tex]\[
x = -\frac{13}{15}
\][/tex]

Thus, the solutions to the equation [tex]\(15x^2 + 13x = 0\)[/tex] are [tex]\(x = 0\)[/tex] and [tex]\(x = -\frac{13}{15}\)[/tex]. Therefore, the correct answer is option a: [tex]\(x = -\frac{13}{15}, 0\)[/tex].