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Answer :
To solve the quadratic equation [tex]\( 15x^2 + 13x = 0 \)[/tex] using the quadratic formula, we start by noting the structure of the quadratic formula, which is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, the equation takes the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex], where:
- [tex]\( a = 15 \)[/tex]
- [tex]\( b = 13 \)[/tex]
- [tex]\( c = 0 \)[/tex]
Since [tex]\( c = 0 \)[/tex], the equation becomes [tex]\( 15x^2 + 13x = 0 \)[/tex]. You can factor this equation as [tex]\( x(15x + 13) = 0 \)[/tex].
This gives two possible solutions:
1. From [tex]\( x = 0 \)[/tex].
2. From [tex]\( 15x + 13 = 0 \)[/tex]:
- To solve this, we set [tex]\( 15x + 13 = 0 \)[/tex].
- Solving for [tex]\( x \)[/tex], we get [tex]\( 15x = -13 \)[/tex].
- Thus, [tex]\( x = -\frac{13}{15} \)[/tex].
So, the solutions to the equation are [tex]\( x = 0 \)[/tex] and [tex]\( x = -\frac{13}{15} \)[/tex].
Looking at the answer choices, option A: [tex]\( x = -\frac{13}{15}, 0 \)[/tex] corresponds to these solutions.
Therefore, the best answer is A.
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, the equation takes the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex], where:
- [tex]\( a = 15 \)[/tex]
- [tex]\( b = 13 \)[/tex]
- [tex]\( c = 0 \)[/tex]
Since [tex]\( c = 0 \)[/tex], the equation becomes [tex]\( 15x^2 + 13x = 0 \)[/tex]. You can factor this equation as [tex]\( x(15x + 13) = 0 \)[/tex].
This gives two possible solutions:
1. From [tex]\( x = 0 \)[/tex].
2. From [tex]\( 15x + 13 = 0 \)[/tex]:
- To solve this, we set [tex]\( 15x + 13 = 0 \)[/tex].
- Solving for [tex]\( x \)[/tex], we get [tex]\( 15x = -13 \)[/tex].
- Thus, [tex]\( x = -\frac{13}{15} \)[/tex].
So, the solutions to the equation are [tex]\( x = 0 \)[/tex] and [tex]\( x = -\frac{13}{15} \)[/tex].
Looking at the answer choices, option A: [tex]\( x = -\frac{13}{15}, 0 \)[/tex] corresponds to these solutions.
Therefore, the best answer is A.
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