Answer :
To solve the equation [tex]\(15x^2 + 13x = 0\)[/tex] using the quadratic formula, we begin by identifying the coefficients of the equation, which has the form [tex]\(ax^2 + bx + c = 0\)[/tex]. In this case, [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]
1. Calculate the Discriminant:
The discriminant is given by [tex]\(b^2 - 4ac\)[/tex].
[tex]\[
b^2 - 4ac = 13^2 - 4 \times 15 \times 0 = 169 - 0 = 169
\][/tex]
2. Calculate the Solutions:
Since the discriminant is positive, we have two real solutions:
[tex]\[
x_1 = \frac{{-b + \sqrt{{169}}}}{{2a}} = \frac{{-13 + 13}}{{2 \times 15}} = \frac{0}{30} = 0
\][/tex]
[tex]\[
x_2 = \frac{{-b - \sqrt{{169}}}}{{2a}} = \frac{{-13 - 13}}{{2 \times 15}} = \frac{{-26}}{{30}} = -\frac{13}{15}
\][/tex]
Therefore, the solutions to the equation are [tex]\(x = 0\)[/tex] and [tex]\(x = -\frac{13}{15}\)[/tex].
Given the options, the correct answer is A: [tex]\(x = -\frac{13}{15}, 0\)[/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]\[
b^2 - 4ac = 13^2 - 4 \times 15 \times 0 = 169 - 0 = 169
\][/tex]
2. Calculate the Solutions:
Since the discriminant is positive, we have two real solutions:
[tex]\[
x_1 = \frac{{-b + \sqrt{{169}}}}{{2a}} = \frac{{-13 + 13}}{{2 \times 15}} = \frac{0}{30} = 0
\][/tex]
[tex]\[
x_2 = \frac{{-b - \sqrt{{169}}}}{{2a}} = \frac{{-13 - 13}}{{2 \times 15}} = \frac{{-26}}{{30}} = -\frac{13}{15}
\][/tex]
Therefore, the solutions to the equation are [tex]\(x = 0\)[/tex] and [tex]\(x = -\frac{13}{15}\)[/tex].
Given the options, the correct answer is A: [tex]\(x = -\frac{13}{15}, 0\)[/tex].
