Answer :
To solve the equation [tex]\(25x^3 - x = 0\)[/tex], we can follow these steps:
1. Factor the Equation:
First, notice that we can factor out an [tex]\(x\)[/tex] from the equation:
[tex]\[
x(25x^2 - 1) = 0
\][/tex]
This gives us two possibilities to consider: [tex]\(x = 0\)[/tex] and [tex]\(25x^2 - 1 = 0\)[/tex].
2. Solve for [tex]\(x = 0\)[/tex]:
The first solution is straightforward:
[tex]\[
x = 0
\][/tex]
3. Solve for [tex]\(25x^2 - 1 = 0\)[/tex]:
Set the quadratic expression equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[
25x^2 - 1 = 0
\][/tex]
[tex]\[
25x^2 = 1
\][/tex]
[tex]\[
x^2 = \frac{1}{25}
\][/tex]
[tex]\[
x = \pm \frac{1}{5}
\][/tex]
4. Conclude the Solutions:
From the factored equation, we found three solutions for [tex]\(x\)[/tex]:
[tex]\[
x = 0, \quad x = \frac{1}{5}, \quad \text{and} \quad x = -\frac{1}{5}
\][/tex]
So, the complete set of solutions for the equation [tex]\(25x^3 - x = 0\)[/tex] is [tex]\(x = 0\)[/tex], [tex]\(x = \frac{1}{5}\)[/tex], and [tex]\(x = -\frac{1}{5}\)[/tex].
1. Factor the Equation:
First, notice that we can factor out an [tex]\(x\)[/tex] from the equation:
[tex]\[
x(25x^2 - 1) = 0
\][/tex]
This gives us two possibilities to consider: [tex]\(x = 0\)[/tex] and [tex]\(25x^2 - 1 = 0\)[/tex].
2. Solve for [tex]\(x = 0\)[/tex]:
The first solution is straightforward:
[tex]\[
x = 0
\][/tex]
3. Solve for [tex]\(25x^2 - 1 = 0\)[/tex]:
Set the quadratic expression equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[
25x^2 - 1 = 0
\][/tex]
[tex]\[
25x^2 = 1
\][/tex]
[tex]\[
x^2 = \frac{1}{25}
\][/tex]
[tex]\[
x = \pm \frac{1}{5}
\][/tex]
4. Conclude the Solutions:
From the factored equation, we found three solutions for [tex]\(x\)[/tex]:
[tex]\[
x = 0, \quad x = \frac{1}{5}, \quad \text{and} \quad x = -\frac{1}{5}
\][/tex]
So, the complete set of solutions for the equation [tex]\(25x^3 - x = 0\)[/tex] is [tex]\(x = 0\)[/tex], [tex]\(x = \frac{1}{5}\)[/tex], and [tex]\(x = -\frac{1}{5}\)[/tex].
