Answer :
To solve the equation [tex]\(15x^2 + 13x = 0\)[/tex] using the quadratic formula, we need to identify the coefficients and apply them to the formula:
The quadratic formula is:
[tex]\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\][/tex]
For the equation [tex]\(15x^2 + 13x = 0\)[/tex], the coefficients are:
- [tex]\(a = 15\)[/tex]
- [tex]\(b = 13\)[/tex]
- [tex]\(c = 0\)[/tex]
Substitute these values into the quadratic formula:
1. Calculate the discriminant [tex]\(b^2 - 4ac\)[/tex]:
[tex]\[
b^2 - 4ac = 13^2 - 4 \times 15 \times 0 = 169
\][/tex]
2. Since the discriminant is 169, we proceed with finding the roots:
[tex]\[
x = \frac{{-13 \pm \sqrt{169}}}{30}
\][/tex]
3. Calculate the square root of the discriminant:
[tex]\[
\sqrt{169} = 13
\][/tex]
4. Substitute back to find the two potential solutions for [tex]\(x\)[/tex]:
- First solution ([tex]\(+\)[/tex]) is:
[tex]\[
x_1 = \frac{{-13 + 13}}{30} = \frac{0}{30} = 0
\][/tex]
- Second solution ([tex]\(-\)[/tex]) is:
[tex]\[
x_2 = \frac{{-13 - 13}}{30} = \frac{-26}{30} = -\frac{13}{15}
\][/tex]
The solutions to the equation are:
- [tex]\(x = 0\)[/tex]
- [tex]\(x = -\frac{13}{15}\)[/tex]
Based on these solutions, the correct answer from the provided 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]
For the equation [tex]\(15x^2 + 13x = 0\)[/tex], the coefficients are:
- [tex]\(a = 15\)[/tex]
- [tex]\(b = 13\)[/tex]
- [tex]\(c = 0\)[/tex]
Substitute these values into the quadratic formula:
1. Calculate the discriminant [tex]\(b^2 - 4ac\)[/tex]:
[tex]\[
b^2 - 4ac = 13^2 - 4 \times 15 \times 0 = 169
\][/tex]
2. Since the discriminant is 169, we proceed with finding the roots:
[tex]\[
x = \frac{{-13 \pm \sqrt{169}}}{30}
\][/tex]
3. Calculate the square root of the discriminant:
[tex]\[
\sqrt{169} = 13
\][/tex]
4. Substitute back to find the two potential solutions for [tex]\(x\)[/tex]:
- First solution ([tex]\(+\)[/tex]) is:
[tex]\[
x_1 = \frac{{-13 + 13}}{30} = \frac{0}{30} = 0
\][/tex]
- Second solution ([tex]\(-\)[/tex]) is:
[tex]\[
x_2 = \frac{{-13 - 13}}{30} = \frac{-26}{30} = -\frac{13}{15}
\][/tex]
The solutions to the equation are:
- [tex]\(x = 0\)[/tex]
- [tex]\(x = -\frac{13}{15}\)[/tex]
Based on these solutions, the correct answer from the provided options is:
A. [tex]\(x = -\frac{13}{15}, 0\)[/tex]
