Answer :
To solve the equation [tex]\(15x^2 + 13x = 0\)[/tex] using the quadratic formula, let's proceed step-by-step.
The given quadratic equation is in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex], where:
- [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]
Let's substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into this formula.
1. Calculate the Discriminant:
[tex]\[
b^2 - 4ac = 13^2 - 4 \cdot 15 \cdot 0 = 169 - 0 = 169
\][/tex]
2. Find the Square Root of the Discriminant:
[tex]\[
\sqrt{169} = 13
\][/tex]
3. Calculate the Two Possible Values for [tex]\(x\)[/tex]:
- First solution ([tex]\(x_1\)[/tex]):
[tex]\[
x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} = \frac{-13 + 13}{2 \cdot 15} = \frac{0}{30} = 0
\][/tex]
- Second solution ([tex]\(x_2\)[/tex]):
[tex]\[
x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a} = \frac{-13 - 13}{2 \cdot 15} = \frac{-26}{30} = -\frac{13}{15}
\][/tex]
Thus, the solutions to the equation [tex]\(15x^2 + 13x = 0\)[/tex] are [tex]\(x = 0\)[/tex] and [tex]\(x = -\frac{13}{15}\)[/tex].
Therefore, the best answer is:
a. [tex]\(x = -\frac{13}{15}, 0\)[/tex]
The given quadratic equation is in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex], where:
- [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]
Let's substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into this formula.
1. Calculate the Discriminant:
[tex]\[
b^2 - 4ac = 13^2 - 4 \cdot 15 \cdot 0 = 169 - 0 = 169
\][/tex]
2. Find the Square Root of the Discriminant:
[tex]\[
\sqrt{169} = 13
\][/tex]
3. Calculate the Two Possible Values for [tex]\(x\)[/tex]:
- First solution ([tex]\(x_1\)[/tex]):
[tex]\[
x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} = \frac{-13 + 13}{2 \cdot 15} = \frac{0}{30} = 0
\][/tex]
- Second solution ([tex]\(x_2\)[/tex]):
[tex]\[
x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a} = \frac{-13 - 13}{2 \cdot 15} = \frac{-26}{30} = -\frac{13}{15}
\][/tex]
Thus, the solutions to the equation [tex]\(15x^2 + 13x = 0\)[/tex] are [tex]\(x = 0\)[/tex] and [tex]\(x = -\frac{13}{15}\)[/tex].
Therefore, the best answer is:
a. [tex]\(x = -\frac{13}{15}, 0\)[/tex]
