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Answer :
To solve the equation [tex]\(15x^2 + 13x = 0\)[/tex] using the quadratic formula, let's follow these steps:
1. Identify the coefficients:
- The equation is in the form [tex]\(ax^2 + bx + c = 0\)[/tex].
- Here, [tex]\(a = 15\)[/tex], [tex]\(b = 13\)[/tex], and [tex]\(c = 0\)[/tex].
2. Quadratic Formula:
- The quadratic formula is [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex].
3. Calculate the discriminant:
- The discriminant is [tex]\(b^2 - 4ac\)[/tex].
- Substitute the values: [tex]\(13^2 - 4 \times 15 \times 0 = 169\)[/tex].
4. Solve for [tex]\(x\)[/tex]:
- Since the discriminant is positive (169), there are two solutions.
- Calculate each solution:
- [tex]\(x_1 = \frac{-13 + \sqrt{169}}{2 \times 15} = \frac{-13 + 13}{30} = \frac{0}{30} = 0\)[/tex].
- [tex]\(x_2 = \frac{-13 - \sqrt{169}}{2 \times 15} = \frac{-13 - 13}{30} = \frac{-26}{30} = -\frac{13}{15}\)[/tex].
5. Conclusion:
- The solutions to the quadratic equation [tex]\(15x^2 + 13x = 0\)[/tex] are [tex]\(x = 0\)[/tex] and [tex]\(x = -\frac{13}{15}\)[/tex].
Among the given choices, the correct answer is A: [tex]\(x = -\frac{13}{15}, 0\)[/tex].
1. Identify the coefficients:
- The equation is in the form [tex]\(ax^2 + bx + c = 0\)[/tex].
- Here, [tex]\(a = 15\)[/tex], [tex]\(b = 13\)[/tex], and [tex]\(c = 0\)[/tex].
2. Quadratic Formula:
- The quadratic formula is [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex].
3. Calculate the discriminant:
- The discriminant is [tex]\(b^2 - 4ac\)[/tex].
- Substitute the values: [tex]\(13^2 - 4 \times 15 \times 0 = 169\)[/tex].
4. Solve for [tex]\(x\)[/tex]:
- Since the discriminant is positive (169), there are two solutions.
- Calculate each solution:
- [tex]\(x_1 = \frac{-13 + \sqrt{169}}{2 \times 15} = \frac{-13 + 13}{30} = \frac{0}{30} = 0\)[/tex].
- [tex]\(x_2 = \frac{-13 - \sqrt{169}}{2 \times 15} = \frac{-13 - 13}{30} = \frac{-26}{30} = -\frac{13}{15}\)[/tex].
5. Conclusion:
- The solutions to the quadratic equation [tex]\(15x^2 + 13x = 0\)[/tex] are [tex]\(x = 0\)[/tex] and [tex]\(x = -\frac{13}{15}\)[/tex].
Among the given choices, the correct answer is A: [tex]\(x = -\frac{13}{15}, 0\)[/tex].
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