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Answer :
To solve the equation [tex]\(15x^2 + 13x = 0\)[/tex], we can use the quadratic formula. The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
First, let's identify the coefficients from the equation [tex]\(15x^2 + 13x = 0\)[/tex]:
- [tex]\(a = 15\)[/tex]
- [tex]\(b = 13\)[/tex]
- [tex]\(c = 0\)[/tex]
Now, we calculate the discriminant, which is the part under the square root in the quadratic formula:
[tex]\[ \text{Discriminant} = b^2 - 4ac = 13^2 - 4 \times 15 \times 0 = 169 \][/tex]
Since the discriminant is 169, which is greater than 0, we will have two real solutions. Now, let's find the roots:
1. First root ([tex]\(x_1\)[/tex]):
[tex]\[ x_1 = \frac{-b + \sqrt{\text{Discriminant}}}{2a} = \frac{-13 + \sqrt{169}}{30} = \frac{-13 + 13}{30} = \frac{0}{30} = 0 \][/tex]
2. Second root ([tex]\(x_2\)[/tex]):
[tex]\[ x_2 = \frac{-b - \sqrt{\text{Discriminant}}}{2a} = \frac{-13 - \sqrt{169}}{30} = \frac{-13 - 13}{30} = \frac{-26}{30} = -\frac{13}{15} \][/tex]
Hence, 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 correct choice from the options provided is:
A. [tex]\(x = -\frac{13}{15}, 0\)[/tex]
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
First, let's identify the coefficients from the equation [tex]\(15x^2 + 13x = 0\)[/tex]:
- [tex]\(a = 15\)[/tex]
- [tex]\(b = 13\)[/tex]
- [tex]\(c = 0\)[/tex]
Now, we calculate the discriminant, which is the part under the square root in the quadratic formula:
[tex]\[ \text{Discriminant} = b^2 - 4ac = 13^2 - 4 \times 15 \times 0 = 169 \][/tex]
Since the discriminant is 169, which is greater than 0, we will have two real solutions. Now, let's find the roots:
1. First root ([tex]\(x_1\)[/tex]):
[tex]\[ x_1 = \frac{-b + \sqrt{\text{Discriminant}}}{2a} = \frac{-13 + \sqrt{169}}{30} = \frac{-13 + 13}{30} = \frac{0}{30} = 0 \][/tex]
2. Second root ([tex]\(x_2\)[/tex]):
[tex]\[ x_2 = \frac{-b - \sqrt{\text{Discriminant}}}{2a} = \frac{-13 - \sqrt{169}}{30} = \frac{-13 - 13}{30} = \frac{-26}{30} = -\frac{13}{15} \][/tex]
Hence, 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 correct choice from the options provided is:
A. [tex]\(x = -\frac{13}{15}, 0\)[/tex]
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