(x+5)^4-10(x+5)^2+9=0

2 min read Jun 17, 2024

Solving the Equation (x+5)^4 - 10(x+5)^2 + 9 = 0

This equation may look complex at first glance, but it can be solved using a clever substitution.

The equation features a pattern: both terms with 'x' are raised to even powers. This suggests a substitution to simplify the equation.

Let's introduce a new variable: y = (x+5)^2.

Now, we can rewrite the equation in terms of 'y':

y^2 - 10y + 9 = 0

This is a simple quadratic equation, which we can solve using various methods, like factoring or the quadratic formula.

In this case, the quadratic equation factors easily:

(y - 9)(y - 1) = 0

This gives us two solutions for 'y':

Now, we need to substitute back to find the values of 'x'. Remember, y = (x+5)^2.

Case 1: y = 9

(x+5)^2 = 9

Taking the square root of both sides:

x + 5 = ±3

Solving for 'x':

Case 2: y = 1

(x+5)^2 = 1

Taking the square root of both sides:

x + 5 = ±1

Solving for 'x':

Therefore, the solutions to the equation (x+5)^4 - 10(x+5)^2 + 9 = 0 are: