Answer :
To solve the equation [tex]\(15x^2 + 13x = 0\)[/tex] using the quadratic formula, let's first identify the coefficients: [tex]\(a = 15\)[/tex], [tex]\(b = 13\)[/tex], and [tex]\(c = 0\)[/tex].
The quadratic formula is:
[tex]\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\][/tex]
Let's go through the steps:
1. Calculate the Discriminant:
[tex]\[
b^2 - 4ac = 13^2 - 4 \cdot 15 \cdot 0 = 169
\][/tex]
2. Apply the Quadratic Formula:
Since the discriminant is 169, which is non-negative, we have real solutions:
[tex]\[
x_1 = \frac{{-13 + \sqrt{169}}}{30} = \frac{{-13 + 13}}{30} = \frac{0}{30} = 0
\][/tex]
[tex]\[
x_2 = \frac{{-13 - \sqrt{169}}}{30} = \frac{{-13 - 13}}{30} = \frac{-26}{30} = -\frac{13}{15}
\][/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].
The correct choice from the given options is:
a. [tex]\(x = -\frac{13}{15}, 0\)[/tex]
The quadratic formula is:
[tex]\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\][/tex]
Let's go through the steps:
1. Calculate the Discriminant:
[tex]\[
b^2 - 4ac = 13^2 - 4 \cdot 15 \cdot 0 = 169
\][/tex]
2. Apply the Quadratic Formula:
Since the discriminant is 169, which is non-negative, we have real solutions:
[tex]\[
x_1 = \frac{{-13 + \sqrt{169}}}{30} = \frac{{-13 + 13}}{30} = \frac{0}{30} = 0
\][/tex]
[tex]\[
x_2 = \frac{{-13 - \sqrt{169}}}{30} = \frac{{-13 - 13}}{30} = \frac{-26}{30} = -\frac{13}{15}
\][/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].
The correct choice from the given options is:
a. [tex]\(x = -\frac{13}{15}, 0\)[/tex]
