Answer :
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]
- [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]
