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Answer :
To solve the equation [tex]\(15x^2 + 13x = 0\)[/tex] using the quadratic formula, we can start by identifying the coefficients. The equation is in the form [tex]\( ax^2 + bx + c = 0 \)[/tex], where:
- [tex]\( a = 15 \)[/tex]
- [tex]\( b = 13 \)[/tex]
- [tex]\( c = 0 \)[/tex]
The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here's how you solve the equation step-by-step:
1. Calculate the Discriminant:
The discriminant in the quadratic formula is given by:
[tex]\[
b^2 - 4ac
\][/tex]
Plugging in the values:
[tex]\[
13^2 - 4 \times 15 \times 0 = 169
\][/tex]
The discriminant is 169.
2. Calculate the Solutions:
The solutions for [tex]\( x \)[/tex] are found by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Substituting the known values:
[tex]\[
x = \frac{-13 \pm \sqrt{169}}{2 \times 15}
\][/tex]
Simplifying further:
[tex]\[
x = \frac{-13 \pm 13}{30}
\][/tex]
This gives us two possible solutions:
- When the "+" sign is used:
[tex]\[
x = \frac{-13 + 13}{30} = \frac{0}{30} = 0
\][/tex]
- When the "−" sign is used:
[tex]\[
x = \frac{-13 - 13}{30} = \frac{-26}{30} = -\frac{13}{15}
\][/tex]
Thus, the solutions to the equation are [tex]\( x = 0 \)[/tex] and [tex]\( x = -\frac{13}{15} \)[/tex].
The correct choice among the given options 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:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here's how you solve the equation step-by-step:
1. Calculate the Discriminant:
The discriminant in the quadratic formula is given by:
[tex]\[
b^2 - 4ac
\][/tex]
Plugging in the values:
[tex]\[
13^2 - 4 \times 15 \times 0 = 169
\][/tex]
The discriminant is 169.
2. Calculate the Solutions:
The solutions for [tex]\( x \)[/tex] are found by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Substituting the known values:
[tex]\[
x = \frac{-13 \pm \sqrt{169}}{2 \times 15}
\][/tex]
Simplifying further:
[tex]\[
x = \frac{-13 \pm 13}{30}
\][/tex]
This gives us two possible solutions:
- When the "+" sign is used:
[tex]\[
x = \frac{-13 + 13}{30} = \frac{0}{30} = 0
\][/tex]
- When the "−" sign is used:
[tex]\[
x = \frac{-13 - 13}{30} = \frac{-26}{30} = -\frac{13}{15}
\][/tex]
Thus, the solutions to the equation are [tex]\( x = 0 \)[/tex] and [tex]\( x = -\frac{13}{15} \)[/tex].
The correct choice among the given options is:
a. [tex]\( x = -\frac{13}{15}, 0 \)[/tex]
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