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Answer :
To solve the equation [tex]\(25x^3 = 64x\)[/tex], we can follow these steps:
1. Rearrange the Equation: Start by moving all terms to one side of the equation to set it to zero:
[tex]\[
25x^3 - 64x = 0
\][/tex]
2. Factor Out the Common Term: Notice that [tex]\(x\)[/tex] is a common factor in both terms. Factor out [tex]\(x\)[/tex]:
[tex]\[
x(25x^2 - 64) = 0
\][/tex]
3. Solve for Zero: When a product of factors is equal to zero, at least one of the factors must be zero. This gives us two separate equations to solve:
- [tex]\(x = 0\)[/tex]
- [tex]\(25x^2 - 64 = 0\)[/tex]
4. Solve the Quadratic Equation: For [tex]\(25x^2 - 64 = 0\)[/tex], add 64 to both sides:
[tex]\[
25x^2 = 64
\][/tex]
Divide both sides by 25:
[tex]\[
x^2 = \frac{64}{25}
\][/tex]
5. Take the Square Root: Solve for [tex]\(x\)[/tex] by taking the square root of both sides. Remember to consider both the positive and negative roots:
[tex]\[
x = \pm \frac{8}{5}
\][/tex]
6. List All Solutions: The full set of solutions for the equation [tex]\(25x^3 = 64x\)[/tex] is:
[tex]\[
x = 0, \quad x = \frac{8}{5}, \quad x = -\frac{8}{5}
\][/tex]
These are the solutions to the equation based on the steps above.
1. Rearrange the Equation: Start by moving all terms to one side of the equation to set it to zero:
[tex]\[
25x^3 - 64x = 0
\][/tex]
2. Factor Out the Common Term: Notice that [tex]\(x\)[/tex] is a common factor in both terms. Factor out [tex]\(x\)[/tex]:
[tex]\[
x(25x^2 - 64) = 0
\][/tex]
3. Solve for Zero: When a product of factors is equal to zero, at least one of the factors must be zero. This gives us two separate equations to solve:
- [tex]\(x = 0\)[/tex]
- [tex]\(25x^2 - 64 = 0\)[/tex]
4. Solve the Quadratic Equation: For [tex]\(25x^2 - 64 = 0\)[/tex], add 64 to both sides:
[tex]\[
25x^2 = 64
\][/tex]
Divide both sides by 25:
[tex]\[
x^2 = \frac{64}{25}
\][/tex]
5. Take the Square Root: Solve for [tex]\(x\)[/tex] by taking the square root of both sides. Remember to consider both the positive and negative roots:
[tex]\[
x = \pm \frac{8}{5}
\][/tex]
6. List All Solutions: The full set of solutions for the equation [tex]\(25x^3 = 64x\)[/tex] is:
[tex]\[
x = 0, \quad x = \frac{8}{5}, \quad x = -\frac{8}{5}
\][/tex]
These are the solutions to the equation based on the steps above.
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