Solve the equation: [tex]p^3-100 p=0[/tex]
Answer (2)
Factor out p from the equation: p ( p 2 − 100 ) = 0 .
Factor the quadratic term using the difference of squares: p ( p − 10 ) ( p + 10 ) = 0 .
Set each factor equal to zero: p = 0 , p − 10 = 0 , p + 10 = 0 .
Solve for p : p = 0 , p = 10 , p = − 10 . Thus, p = 0 , 10 , − 10 .
Explanation
Understanding the Problem We are given the equation p 3 − 100 p = 0 . Our goal is to find all possible values of p that satisfy this equation.
Factoring out p First, we can factor out a common factor of p from both terms on the left side of the equation: p ( p 2 − 100 ) = 0
Factoring the Difference of Squares Now, we recognize that the term p 2 − 100 is a difference of squares, which can be factored as ( p − 10 ) ( p + 10 ) . So, we have: p ( p − 10 ) ( p + 10 ) = 0
Solving for p To find the solutions for p , we set each factor equal to zero:
p = 0
p − 10 = 0 ⇒ p = 10
p + 10 = 0 ⇒ p = − 10
Thus, the solutions are p = 0 , p = 10 , and p = − 10 .
Final Answer Therefore, the solutions to the equation p 3 − 100 p = 0 are p = 0 , 10 , − 10 .
Examples
Imagine you are designing a garden and want to know the possible dimensions for a rectangular section with a specific area. If the area is represented by the equation x 3 − 9 x = 0 , where x is a dimension, solving this equation will give you the possible values for that dimension. This type of problem arises in various fields, including engineering, physics, and economics, where equations need to be solved to find possible values for variables in real-world scenarios.
The equation p 3 − 100 p = 0 can be solved by factoring out p and using the difference of squares. The solutions are p = 0 , p = 10 , and p = − 10 . Therefore, the complete set of solutions is p = 0 , 10 , − 10 .
;
