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Answer :
To solve the quadratic equation [tex]\(15x^2 + 13x = 0\)[/tex] using the quadratic formula, we follow these steps:
1. Identify the coefficients: In the equation [tex]\(15x^2 + 13x = 0\)[/tex], we recognize it as a quadratic equation of the form [tex]\(ax^2 + bx + c = 0\)[/tex]. Here, [tex]\(a = 15\)[/tex], [tex]\(b = 13\)[/tex], and [tex]\(c = 0\)[/tex].
2. Use the 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 given by [tex]\(b^2 - 4ac\)[/tex].
[tex]\[
\Delta = 13^2 - 4 \times 15 \times 0 = 169
\][/tex]
4. Compute the roots: Since the discriminant is positive, the equation has two real roots.
[tex]\[
x_1 = \frac{-b + \sqrt{\Delta}}{2a} = \frac{-13 + \sqrt{169}}{30}
\][/tex]
[tex]\[
x_2 = \frac{-b - \sqrt{\Delta}}{2a} = \frac{-13 - \sqrt{169}}{30}
\][/tex]
Calculating these:
[tex]\[
x_1 = \frac{-13 + 13}{30} = \frac{0}{30} = 0
\][/tex]
[tex]\[
x_2 = \frac{-13 - 13}{30} = \frac{-26}{30} = -\frac{13}{15}
\][/tex]
5. Solution: The roots of the equation [tex]\(15x^2 + 13x = 0\)[/tex] are [tex]\(x = 0\)[/tex] and [tex]\(x = -\frac{13}{15}\)[/tex].
Therefore, the correct choice is:
- [tex]\(x = 0, -\frac{13}{15}\)[/tex]
This corresponds to option C in the original problem.
1. Identify the coefficients: In the equation [tex]\(15x^2 + 13x = 0\)[/tex], we recognize it as a quadratic equation of the form [tex]\(ax^2 + bx + c = 0\)[/tex]. Here, [tex]\(a = 15\)[/tex], [tex]\(b = 13\)[/tex], and [tex]\(c = 0\)[/tex].
2. Use the 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 given by [tex]\(b^2 - 4ac\)[/tex].
[tex]\[
\Delta = 13^2 - 4 \times 15 \times 0 = 169
\][/tex]
4. Compute the roots: Since the discriminant is positive, the equation has two real roots.
[tex]\[
x_1 = \frac{-b + \sqrt{\Delta}}{2a} = \frac{-13 + \sqrt{169}}{30}
\][/tex]
[tex]\[
x_2 = \frac{-b - \sqrt{\Delta}}{2a} = \frac{-13 - \sqrt{169}}{30}
\][/tex]
Calculating these:
[tex]\[
x_1 = \frac{-13 + 13}{30} = \frac{0}{30} = 0
\][/tex]
[tex]\[
x_2 = \frac{-13 - 13}{30} = \frac{-26}{30} = -\frac{13}{15}
\][/tex]
5. Solution: The roots of the equation [tex]\(15x^2 + 13x = 0\)[/tex] are [tex]\(x = 0\)[/tex] and [tex]\(x = -\frac{13}{15}\)[/tex].
Therefore, the correct choice is:
- [tex]\(x = 0, -\frac{13}{15}\)[/tex]
This corresponds to option C in the original problem.
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