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Answer :
To solve the equation [tex]\(15x^2 + 13x = 0\)[/tex] using the quadratic formula, let's break it down step by step.
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
First, 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, calculate the discriminant [tex]\(\Delta = b^2 - 4ac\)[/tex]:
[tex]\[ \Delta = 13^2 - 4 \times 15 \times 0 \][/tex]
[tex]\[ \Delta = 169 \][/tex]
Since [tex]\( \Delta = 169 \)[/tex], which is positive, we know that there are two real solutions. Next, apply the quadratic formula:
1. Calculate the first solution [tex]\(x_1\)[/tex]:
[tex]\[ x_1 = \frac{-13 + \sqrt{169}}{2 \times 15} \][/tex]
Since [tex]\(\sqrt{169} = 13\)[/tex]:
[tex]\[ x_1 = \frac{-13 + 13}{30} = \frac{0}{30} = 0 \][/tex]
2. Calculate the second solution [tex]\(x_2\)[/tex]:
[tex]\[ x_2 = \frac{-13 - \sqrt{169}}{2 \times 15} \][/tex]
Again, since [tex]\(\sqrt{169} = 13\)[/tex]:
[tex]\[ x_2 = \frac{-13 - 13}{30} = \frac{-26}{30} = -\frac{13}{15} \][/tex]
So, the solutions to the equation [tex]\(15x^2 + 13x = 0\)[/tex] are [tex]\(x = 0\)[/tex] and [tex]\(x = -\frac{13}{15}\)[/tex].
From the choices provided, the correct answer is:
A: [tex]\(x = -\frac{13}{15}, 0\)[/tex]
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
First, 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, calculate the discriminant [tex]\(\Delta = b^2 - 4ac\)[/tex]:
[tex]\[ \Delta = 13^2 - 4 \times 15 \times 0 \][/tex]
[tex]\[ \Delta = 169 \][/tex]
Since [tex]\( \Delta = 169 \)[/tex], which is positive, we know that there are two real solutions. Next, apply the quadratic formula:
1. Calculate the first solution [tex]\(x_1\)[/tex]:
[tex]\[ x_1 = \frac{-13 + \sqrt{169}}{2 \times 15} \][/tex]
Since [tex]\(\sqrt{169} = 13\)[/tex]:
[tex]\[ x_1 = \frac{-13 + 13}{30} = \frac{0}{30} = 0 \][/tex]
2. Calculate the second solution [tex]\(x_2\)[/tex]:
[tex]\[ x_2 = \frac{-13 - \sqrt{169}}{2 \times 15} \][/tex]
Again, since [tex]\(\sqrt{169} = 13\)[/tex]:
[tex]\[ x_2 = \frac{-13 - 13}{30} = \frac{-26}{30} = -\frac{13}{15} \][/tex]
So, the solutions to the equation [tex]\(15x^2 + 13x = 0\)[/tex] are [tex]\(x = 0\)[/tex] and [tex]\(x = -\frac{13}{15}\)[/tex].
From the choices provided, the correct answer is:
A: [tex]\(x = -\frac{13}{15}, 0\)[/tex]
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