Answer :
To solve the equation [tex]\(25x^3 - x = 0\)[/tex], we can follow these steps:
1. Factor the equation: Start by factoring out the common term, [tex]\(x\)[/tex], from the equation:
[tex]\[
x(25x^2 - 1) = 0
\][/tex]
2. Set each factor to zero: The equation is satisfied when any of the factors equals zero.
- For the factor [tex]\(x = 0\)[/tex], we already have one solution: [tex]\(x = 0\)[/tex].
- For the factor [tex]\(25x^2 - 1 = 0\)[/tex], we need to solve this quadratic equation.
3. Solve the quadratic equation:
[tex]\[
25x^2 - 1 = 0
\][/tex]
Add 1 to both sides:
[tex]\[
25x^2 = 1
\][/tex]
Divide both sides by 25:
[tex]\[
x^2 = \frac{1}{25}
\][/tex]
Take the square root of both sides:
[tex]\[
x = \pm \frac{1}{5}
\][/tex]
4. Combine all solutions: We have three solutions from the above steps:
- [tex]\(x = 0\)[/tex]
- [tex]\(x = \frac{1}{5}\)[/tex]
- [tex]\(x = -\frac{1}{5}\)[/tex]
Therefore, the solutions to the equation [tex]\(25x^3 - x = 0\)[/tex] are [tex]\(-\frac{1}{5}\)[/tex], [tex]\(0\)[/tex], and [tex]\(\frac{1}{5}\)[/tex].
1. Factor the equation: Start by factoring out the common term, [tex]\(x\)[/tex], from the equation:
[tex]\[
x(25x^2 - 1) = 0
\][/tex]
2. Set each factor to zero: The equation is satisfied when any of the factors equals zero.
- For the factor [tex]\(x = 0\)[/tex], we already have one solution: [tex]\(x = 0\)[/tex].
- For the factor [tex]\(25x^2 - 1 = 0\)[/tex], we need to solve this quadratic equation.
3. Solve the quadratic equation:
[tex]\[
25x^2 - 1 = 0
\][/tex]
Add 1 to both sides:
[tex]\[
25x^2 = 1
\][/tex]
Divide both sides by 25:
[tex]\[
x^2 = \frac{1}{25}
\][/tex]
Take the square root of both sides:
[tex]\[
x = \pm \frac{1}{5}
\][/tex]
4. Combine all solutions: We have three solutions from the above steps:
- [tex]\(x = 0\)[/tex]
- [tex]\(x = \frac{1}{5}\)[/tex]
- [tex]\(x = -\frac{1}{5}\)[/tex]
Therefore, the solutions to the equation [tex]\(25x^3 - x = 0\)[/tex] are [tex]\(-\frac{1}{5}\)[/tex], [tex]\(0\)[/tex], and [tex]\(\frac{1}{5}\)[/tex].
