Solve for x.
x³ + ax² - a²x - a³ = 0
(x+a)^2(x-a)
To solve the equation x³ + ax² - a²x - a³ = 0, we can use a method called factoring by grouping. Here's how it works:
Step 1: Group the terms
Group the terms in pairs using parentheses, based on their common factors:
(x³ + ax²) - (a²x + a³) = 0
Step 2: Factor out common terms
Factor out the common terms from each group:
x²(x + a) - a²(x + a) = 0
Step 3: Factor out the common binomial
Factor out the common binomial (x + a):
(x + a)(x² - a²) = 0
Step 4: Factor the difference of squares
Use the difference of squares formula to factor x² - a²:
(x + a)(x + a)(x - a) = 0
Step 5: Set each factor to zero
Set each factor equal to zero and solve for x:
x + a = 0 --> x = -a
x + a = 0 --> x = -a
x - a = 0 --> x = a
Therefore, the solutions to the equation x³ + ax² - a²x - a³ = 0 are x = -a, x = -a, and x = a.
To solve the equation x³ + ax² - a²x - a³ = 0, we can use a method called factoring by grouping. Here's how to do it:
Step 1: Group the terms.
Rearrange the equation by grouping the terms with common factors:
x³ + ax² - a²x - a³ = 0
(x³ + ax²) - (a²x + a³) = 0
Step 2: Factor out the common factors.
Factor out common terms from each group:
x²(x + a) - a²(x + a) = 0
Step 3: Factor out the common binomial.
Notice that (x + a) is a common factor that can be factored out:
(x + a)(x² - a²) = 0
Step 4: Factor the difference of squares.
The expression x² - a² is a difference of squares, which can be factored as (x - a)(x + a):
(x + a)(x - a)(x + a) = 0
Step 5: Set each factor equal to zero.
To solve for x, set each factor equal to zero and solve for x:
x + a = 0 --> x = -a
x - a = 0 --> x = a
x + a = 0 --> x = -a
Therefore, the solutions to the equation x³ + ax² - a²x - a³ = 0 are x = -a, x = a, and x = -a.