9x^5 - 72x^2 = 0
Understand the Problem
The question is asking us to solve the equation 9x^5 - 72x^2 = 0 for the value of x. This involves factoring out common terms and finding the roots of the equation.
Answer
$x = 0$ and $x = 2$
Answer for screen readers
The solutions to the equation are $x = 0$ and $x = 2$.
Steps to Solve
- Identify the common factor
We start by factoring out the common term from the equation $9x^5 - 72x^2 = 0$.
The common factor here is $9x^2$.
So we can rewrite the equation as: $$ 9x^2(x^3 - 8) = 0 $$
- Set each factor to zero
Now we set each factor equal to zero:
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$9x^2 = 0$
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$x^3 - 8 = 0$
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Solve the first equation
For the first equation, $9x^2 = 0$, we divide both sides by 9:
$$ x^2 = 0 $$
Taking the square root of both sides gives us:
$$ x = 0 $$
- Solve the second equation
Now we solve the second equation, $x^3 - 8 = 0$.
We can add 8 to both sides: $$ x^3 = 8 $$
Then take the cube root of both sides: $$ x = \sqrt[3]{8} $$
This simplifies to: $$ x = 2 $$
- List all solutions
From both parts, we found the solutions to the original equation are:
- $x = 0$
- $x = 2$
The solutions to the equation are $x = 0$ and $x = 2$.
More Information
When solving polynomial equations, factoring out the greatest common factor simplifies the equation and makes it easier to find the roots. Here we utilized both square roots and cube roots to find the solutions.
Tips
- Misidentifying the greatest common factor when factoring.
- Forgetting to set each factor equal to zero, which can lead to missing some solutions.
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