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Answer :
To solve the equation [tex]\(15x^2 + 13x = 0\)[/tex] using the quadratic formula, we will start by identifying the coefficients:
- [tex]\(a = 15\)[/tex]
- [tex]\(b = 13\)[/tex]
- [tex]\(c = 0\)[/tex]
The quadratic formula is given by:
[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
Let's calculate each component step-by-step:
1. Calculate the Discriminant:
[tex]\[
b^2 - 4ac = 13^2 - 4 \times 15 \times 0 = 169
\][/tex]
2. Evaluate the Solutions:
Since the discriminant is 169 (a positive number), we will have two different real solutions.
- First solution ([tex]\(x_1\)[/tex]):
[tex]\[
x_1 = \frac{{-13 + \sqrt{169}}}{30}
\][/tex]
[tex]\[
x_1 = \frac{{-13 + 13}}{30} = 0
\][/tex]
- Second solution ([tex]\(x_2\)[/tex]):
[tex]\[
x_2 = \frac{{-13 - \sqrt{169}}}{30}
\][/tex]
[tex]\[
x_2 = \frac{{-13 - 13}}{30} = \frac{-26}{30} = -\frac{13}{15}
\][/tex]
Thus, the solutions to the equation are [tex]\(x = 0\)[/tex] and [tex]\(x = -\frac{13}{15}\)[/tex].
Thus, the correct answer is:
a. [tex]\(x = -\frac{13}{15}, 0\)[/tex]
- [tex]\(a = 15\)[/tex]
- [tex]\(b = 13\)[/tex]
- [tex]\(c = 0\)[/tex]
The quadratic formula is given by:
[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
Let's calculate each component step-by-step:
1. Calculate the Discriminant:
[tex]\[
b^2 - 4ac = 13^2 - 4 \times 15 \times 0 = 169
\][/tex]
2. Evaluate the Solutions:
Since the discriminant is 169 (a positive number), we will have two different real solutions.
- First solution ([tex]\(x_1\)[/tex]):
[tex]\[
x_1 = \frac{{-13 + \sqrt{169}}}{30}
\][/tex]
[tex]\[
x_1 = \frac{{-13 + 13}}{30} = 0
\][/tex]
- Second solution ([tex]\(x_2\)[/tex]):
[tex]\[
x_2 = \frac{{-13 - \sqrt{169}}}{30}
\][/tex]
[tex]\[
x_2 = \frac{{-13 - 13}}{30} = \frac{-26}{30} = -\frac{13}{15}
\][/tex]
Thus, the solutions to the equation are [tex]\(x = 0\)[/tex] and [tex]\(x = -\frac{13}{15}\)[/tex].
Thus, the correct answer is:
a. [tex]\(x = -\frac{13}{15}, 0\)[/tex]
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