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Answer :
To solve the equation [tex]\(15x^2 + 13x = 0\)[/tex] using the quadratic formula, let's follow these steps:
The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
In the equation [tex]\(15x^2 + 13x = 0\)[/tex], the coefficients are [tex]\(a = 15\)[/tex], [tex]\(b = 13\)[/tex], and [tex]\(c = 0\)[/tex].
1. Calculate the discriminant:
[tex]\[
b^2 - 4ac = 13^2 - 4 \times 15 \times 0
\][/tex]
[tex]\[
= 169
\][/tex]
2. Calculate the roots using the quadratic formula:
- For the first root ([tex]\(+\)[/tex] sign):
[tex]\[
x_1 = \frac{-13 + \sqrt{169}}{2 \times 15}
\][/tex]
[tex]\[
= \frac{-13 + 13}{30}
\][/tex]
[tex]\[
= \frac{0}{30} = 0
\][/tex]
- For the second root ([tex]\(-\)[/tex] sign):
[tex]\[
x_2 = \frac{-13 - \sqrt{169}}{2 \times 15}
\][/tex]
[tex]\[
= \frac{-13 - 13}{30}
\][/tex]
[tex]\[
= \frac{-26}{30} = -\frac{13}{15}
\][/tex]
Therefore, the solutions are [tex]\(x = 0\)[/tex] and [tex]\(x = -\frac{13}{15}\)[/tex].
So, the best answer choice that matches these solutions is:
a. [tex]\(x = -\frac{13}{15}, 0\)[/tex]
The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
In the equation [tex]\(15x^2 + 13x = 0\)[/tex], the coefficients are [tex]\(a = 15\)[/tex], [tex]\(b = 13\)[/tex], and [tex]\(c = 0\)[/tex].
1. Calculate the discriminant:
[tex]\[
b^2 - 4ac = 13^2 - 4 \times 15 \times 0
\][/tex]
[tex]\[
= 169
\][/tex]
2. Calculate the roots using the quadratic formula:
- For the first root ([tex]\(+\)[/tex] sign):
[tex]\[
x_1 = \frac{-13 + \sqrt{169}}{2 \times 15}
\][/tex]
[tex]\[
= \frac{-13 + 13}{30}
\][/tex]
[tex]\[
= \frac{0}{30} = 0
\][/tex]
- For the second root ([tex]\(-\)[/tex] sign):
[tex]\[
x_2 = \frac{-13 - \sqrt{169}}{2 \times 15}
\][/tex]
[tex]\[
= \frac{-13 - 13}{30}
\][/tex]
[tex]\[
= \frac{-26}{30} = -\frac{13}{15}
\][/tex]
Therefore, the solutions are [tex]\(x = 0\)[/tex] and [tex]\(x = -\frac{13}{15}\)[/tex].
So, the best answer choice that matches these solutions is:
a. [tex]\(x = -\frac{13}{15}, 0\)[/tex]
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