Answer :
To solve the equation [tex]\(15x^2 + 13x = 0\)[/tex] using the quadratic formula, let's follow these steps:
### Step 1: Identify the coefficients
The given quadratic equation is in the form [tex]\(ax^2 + bx + c = 0\)[/tex]. From the equation [tex]\(15x^2 + 13x = 0\)[/tex]:
- [tex]\(a = 15\)[/tex]
- [tex]\(b = 13\)[/tex]
- [tex]\(c = 0\)[/tex]
### Step 2: Apply the quadratic formula
The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
### Step 3: Calculate the discriminant
The discriminant [tex]\(\Delta\)[/tex] is calculated as:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\Delta = 13^2 - 4 \cdot 15 \cdot 0 = 169
\][/tex]
Since the discriminant is positive, there are two real solutions.
### Step 4: Solve for [tex]\(x\)[/tex]
Now, substitute the values into the quadratic formula:
1. First solution ([tex]\(x_1\)[/tex]):
[tex]\[
x_1 = \frac{-b + \sqrt{\Delta}}{2a} = \frac{-13 + \sqrt{169}}{2 \cdot 15} = \frac{-13 + 13}{30} = \frac{0}{30} = 0
\][/tex]
2. Second solution ([tex]\(x_2\)[/tex]):
[tex]\[
x_2 = \frac{-b - \sqrt{\Delta}}{2a} = \frac{-13 - \sqrt{169}}{2 \cdot 15} = \frac{-13 - 13}{30} = \frac{-26}{30} = -\frac{13}{15}
\][/tex]
### Conclusion:
The solutions to the equation [tex]\(15x^2 + 13x = 0\)[/tex] are [tex]\(x = 0\)[/tex] and [tex]\(x = -\frac{13}{15}\)[/tex].
Thus, the correct option is:
a. [tex]\(x = -\frac{13}{15}, 0\)[/tex]
### Step 1: Identify the coefficients
The given quadratic equation is in the form [tex]\(ax^2 + bx + c = 0\)[/tex]. From the equation [tex]\(15x^2 + 13x = 0\)[/tex]:
- [tex]\(a = 15\)[/tex]
- [tex]\(b = 13\)[/tex]
- [tex]\(c = 0\)[/tex]
### Step 2: Apply the quadratic formula
The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
### Step 3: Calculate the discriminant
The discriminant [tex]\(\Delta\)[/tex] is calculated as:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\Delta = 13^2 - 4 \cdot 15 \cdot 0 = 169
\][/tex]
Since the discriminant is positive, there are two real solutions.
### Step 4: Solve for [tex]\(x\)[/tex]
Now, substitute the values into the quadratic formula:
1. First solution ([tex]\(x_1\)[/tex]):
[tex]\[
x_1 = \frac{-b + \sqrt{\Delta}}{2a} = \frac{-13 + \sqrt{169}}{2 \cdot 15} = \frac{-13 + 13}{30} = \frac{0}{30} = 0
\][/tex]
2. Second solution ([tex]\(x_2\)[/tex]):
[tex]\[
x_2 = \frac{-b - \sqrt{\Delta}}{2a} = \frac{-13 - \sqrt{169}}{2 \cdot 15} = \frac{-13 - 13}{30} = \frac{-26}{30} = -\frac{13}{15}
\][/tex]
### Conclusion:
The solutions to the equation [tex]\(15x^2 + 13x = 0\)[/tex] are [tex]\(x = 0\)[/tex] and [tex]\(x = -\frac{13}{15}\)[/tex].
Thus, the correct option is:
a. [tex]\(x = -\frac{13}{15}, 0\)[/tex]
