Answer :
To solve the equation [tex]\(15x^2 + 13x = 0\)[/tex] using the quadratic formula, we follow these steps:
1. Identify the coefficients: The standard form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex]. Here, [tex]\(a = 15\)[/tex], [tex]\(b = 13\)[/tex], and [tex]\(c = 0\)[/tex].
2. Quadratic Formula: The quadratic formula is given by:
[tex]\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\][/tex]
3. Calculate the discriminant: The discriminant [tex]\(\Delta\)[/tex] is calculated as [tex]\(b^2 - 4ac\)[/tex].
[tex]\[
\Delta = 13^2 - 4 \cdot 15 \cdot 0 = 169
\][/tex]
4. Plug values into the quadratic formula:
[tex]\[
x_1 = \frac{{-13 + \sqrt{169}}}{30}
\][/tex]
[tex]\[
x_2 = \frac{{-13 - \sqrt{169}}}{30}
\][/tex]
5. Simplify:
- For [tex]\(x_1\)[/tex]:
[tex]\[
x_1 = \frac{{-13 + 13}}{30} = \frac{0}{30} = 0
\][/tex]
- For [tex]\(x_2\)[/tex]:
[tex]\[
x_2 = \frac{{-13 - 13}}{30} = \frac{-26}{30} = -\frac{13}{15}
\][/tex]
Thus, the solutions to the equation are [tex]\(x = 0\)[/tex] and [tex]\(x = -\frac{13}{15}\)[/tex].
The correct set of answers corresponding to the given options is:
- b. [tex]\(x=0\)[/tex]
- The other root calculated here matches none of the provided options directly; however, the important solutions are [tex]\(x = 0\)[/tex] and [tex]\(x = -\frac{13}{15}\)[/tex].
1. Identify the coefficients: The standard form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex]. Here, [tex]\(a = 15\)[/tex], [tex]\(b = 13\)[/tex], and [tex]\(c = 0\)[/tex].
2. Quadratic Formula: The quadratic formula is given by:
[tex]\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\][/tex]
3. Calculate the discriminant: The discriminant [tex]\(\Delta\)[/tex] is calculated as [tex]\(b^2 - 4ac\)[/tex].
[tex]\[
\Delta = 13^2 - 4 \cdot 15 \cdot 0 = 169
\][/tex]
4. Plug values into the quadratic formula:
[tex]\[
x_1 = \frac{{-13 + \sqrt{169}}}{30}
\][/tex]
[tex]\[
x_2 = \frac{{-13 - \sqrt{169}}}{30}
\][/tex]
5. Simplify:
- For [tex]\(x_1\)[/tex]:
[tex]\[
x_1 = \frac{{-13 + 13}}{30} = \frac{0}{30} = 0
\][/tex]
- For [tex]\(x_2\)[/tex]:
[tex]\[
x_2 = \frac{{-13 - 13}}{30} = \frac{-26}{30} = -\frac{13}{15}
\][/tex]
Thus, the solutions to the equation are [tex]\(x = 0\)[/tex] and [tex]\(x = -\frac{13}{15}\)[/tex].
The correct set of answers corresponding to the given options is:
- b. [tex]\(x=0\)[/tex]
- The other root calculated here matches none of the provided options directly; however, the important solutions are [tex]\(x = 0\)[/tex] and [tex]\(x = -\frac{13}{15}\)[/tex].
