(x-3)^4-5(x-3)^2-36=0

2 min read Jun 17, 2024

Solving the Equation (x-3)^4 - 5(x-3)^2 - 36 = 0

This equation may seem intimidating at first glance, but it can be solved efficiently using a simple substitution technique. Let's break down the steps:

1. Substitute and Simplify

We can simplify the equation by substituting a new variable. Let y = (x-3). Substituting this into the equation, we get:

y^4 - 5y^2 - 36 = 0

This is now a quadratic equation in terms of 'y'.

2. Factor the Quadratic

The quadratic equation can be factored as follows:

(y^2 - 9)(y^2 + 4) = 0

This gives us two factors:

y^2 - 9 = 0
y^2 + 4 = 0

3. Solve for 'y'

Solving for 'y' in each factor:

y^2 - 9 = 0 => y = ±3
y^2 + 4 = 0 => y = ±2i (where 'i' is the imaginary unit, √-1)

4. Substitute back to find 'x'

Now we substitute back y = (x-3) to find the values of 'x':

y = 3:
- (x-3) = 3
- x = 6
y = -3:
- (x-3) = -3
- x = 0
y = 2i:
- (x-3) = 2i
- x = 3 + 2i
y = -2i:
- (x-3) = -2i
- x = 3 - 2i

Conclusion

Therefore, the solutions to the equation (x-3)^4 - 5(x-3)^2 - 36 = 0 are:

x = 6
x = 0
x = 3 + 2i
x = 3 - 2i