%5E3%3D25(x%2B6).jpeg "(x+6)^3=25(x+6)")
Solving the Equation (x+6)^3 = 25(x+6)
This equation presents a cubic equation with a seemingly complex form. However, we can simplify it by employing a strategic approach. Here's a breakdown of the steps involved in solving the equation:
1. Simplifying the Equation
First, let's simplify the equation by bringing all terms to one side:
(x+6)^3 - 25(x+6) = 0
Now we can see a common factor of (x+6). Let's factor it out:
(x+6) [(x+6)^2 - 25] = 0
2. Applying the Difference of Squares
The expression inside the brackets resembles the difference of squares pattern: a^2 - b^2 = (a+b)(a-b). Applying this, we get:
(x+6) [(x+6) + 5] [(x+6) - 5] = 0
Simplifying further:
(x+6) (x+11) (x+1) = 0
3. Finding the Solutions
Now, for the product of three factors to equal zero, at least one of them must be zero. This gives us three potential solutions:
- x + 6 = 0 => x = -6
- x + 11 = 0 => x = -11
- x + 1 = 0 => x = -1
Therefore, the solutions to the equation (x+6)^3 = 25(x+6) are x = -6, x = -11, and x = -1.