%5E4-13(x%2B3)%5E2%2B36%3D0.jpeg "(x+3)^4-13(x+3)^2+36=0")
Solving the Equation (x+3)^4 - 13(x+3)^2 + 36 = 0
This equation might look intimidating at first glance, but it can be solved using a simple substitution technique.
1. Substitution
Let's introduce a new variable, y = (x+3). This allows us to rewrite the equation as:
y^4 - 13y^2 + 36 = 0
This looks much simpler, right? It's now a quadratic equation in terms of y^2.
2. Factoring the Quadratic
Now we can factor this equation:
(y^2 - 9)(y^2 - 4) = 0
This gives us two possible solutions:
- y^2 - 9 = 0
- y^2 - 4 = 0
3. Solving for y
Solving these equations gives us:
- y^2 = 9 => y = ±3
- y^2 = 4 => y = ±2
4. Back Substitution
Now we need to substitute back x + 3 for y:
- x + 3 = 3 => x = 0
- x + 3 = -3 => x = -6
- x + 3 = 2 => x = -1
- x + 3 = -2 => x = -5
5. Solutions
Therefore, the solutions to the equation (x+3)^4 - 13(x+3)^2 + 36 = 0 are:
x = 0, x = -6, x = -1, x = -5