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Answer :
To solve the equation [tex]\(15x^2 + 13x = 0\)[/tex] using the quadratic formula, we need to correctly identify the roots of the quadratic equation. The standard form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex].
For this equation:
- [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]
In this case, since [tex]\(c = 0\)[/tex], one of the roots of the equation will be zero.
We need to calculate the discriminant, which is [tex]\(b^2 - 4ac\)[/tex]:
[tex]\[ b^2 = 13^2 = 169 \][/tex]
[tex]\[ 4ac = 4 \times 15 \times 0 = 0 \][/tex]
So, the discriminant is [tex]\(169 - 0 = 169\)[/tex], which is positive.
Using the quadratic formula, we find the two roots:
1. First root ([tex]\(x_1\)[/tex]):
[tex]\[
x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} = \frac{-13 + \sqrt{169}}{30} = \frac{-13 + 13}{30} = \frac{0}{30} = 0
\][/tex]
2. Second root ([tex]\(x_2\)[/tex]):
[tex]\[
x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a} = \frac{-13 - \sqrt{169}}{30} = \frac{-13 - 13}{30} = \frac{-26}{30} = -\frac{13}{15}
\][/tex]
Thus, the solutions are [tex]\(x = 0\)[/tex] and [tex]\(x = -\frac{13}{15}\)[/tex].
Therefore, the correct answer is:
a. [tex]\(x = -\frac{13}{15}, 0\)[/tex]
For this equation:
- [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]
In this case, since [tex]\(c = 0\)[/tex], one of the roots of the equation will be zero.
We need to calculate the discriminant, which is [tex]\(b^2 - 4ac\)[/tex]:
[tex]\[ b^2 = 13^2 = 169 \][/tex]
[tex]\[ 4ac = 4 \times 15 \times 0 = 0 \][/tex]
So, the discriminant is [tex]\(169 - 0 = 169\)[/tex], which is positive.
Using the quadratic formula, we find the two roots:
1. First root ([tex]\(x_1\)[/tex]):
[tex]\[
x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} = \frac{-13 + \sqrt{169}}{30} = \frac{-13 + 13}{30} = \frac{0}{30} = 0
\][/tex]
2. Second root ([tex]\(x_2\)[/tex]):
[tex]\[
x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a} = \frac{-13 - \sqrt{169}}{30} = \frac{-13 - 13}{30} = \frac{-26}{30} = -\frac{13}{15}
\][/tex]
Thus, the solutions are [tex]\(x = 0\)[/tex] and [tex]\(x = -\frac{13}{15}\)[/tex].
Therefore, the correct answer is:
a. [tex]\(x = -\frac{13}{15}, 0\)[/tex]
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