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Answer :
We start with the quadratic equation
$$
15x^2 + 13x = 0.
$$
This equation is in the standard form
$$
ax^2 + bx + c = 0,
$$
where
$$
a = 15,\quad b = 13,\quad c = 0.
$$
The quadratic formula is given by
$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
$$
1. First, compute the discriminant:
$$
b^2 - 4ac = 13^2 - 4(15)(0) = 169.
$$
2. Take the square root of the discriminant:
$$
\sqrt{169} = 13.
$$
3. Substitute the values into the quadratic formula:
$$
x = \frac{-13 \pm 13}{2 \cdot 15} = \frac{-13 \pm 13}{30}.
$$
4. This gives us two solutions:
- For the plus sign:
$$
x = \frac{-13 + 13}{30} = \frac{0}{30} = 0.
$$
- For the minus sign:
$$
x = \frac{-13 - 13}{30} = \frac{-26}{30} = -\frac{13}{15}.
$$
Hence, the two solutions are
$$
x = 0 \quad \text{and} \quad x = -\frac{13}{15}.
$$
Looking at the provided answer choices, the correct answer is:
a. $x = -\frac{13}{15},\; 0$.
Thus, the best answer is A.
$$
15x^2 + 13x = 0.
$$
This equation is in the standard form
$$
ax^2 + bx + c = 0,
$$
where
$$
a = 15,\quad b = 13,\quad c = 0.
$$
The quadratic formula is given by
$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
$$
1. First, compute the discriminant:
$$
b^2 - 4ac = 13^2 - 4(15)(0) = 169.
$$
2. Take the square root of the discriminant:
$$
\sqrt{169} = 13.
$$
3. Substitute the values into the quadratic formula:
$$
x = \frac{-13 \pm 13}{2 \cdot 15} = \frac{-13 \pm 13}{30}.
$$
4. This gives us two solutions:
- For the plus sign:
$$
x = \frac{-13 + 13}{30} = \frac{0}{30} = 0.
$$
- For the minus sign:
$$
x = \frac{-13 - 13}{30} = \frac{-26}{30} = -\frac{13}{15}.
$$
Hence, the two solutions are
$$
x = 0 \quad \text{and} \quad x = -\frac{13}{15}.
$$
Looking at the provided answer choices, the correct answer is:
a. $x = -\frac{13}{15},\; 0$.
Thus, the best answer is A.
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