(x+8)^4-3(x+8)^2-28=0

2 min read Jun 17, 2024

Solving the Equation (x+8)^4 - 3(x+8)^2 - 28 = 0

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

Notice that the equation is composed of terms with (x+8) raised to powers of 4 and 2. This suggests a pattern we can exploit.

Let's simplify the equation by making a substitution:

Let y = (x+8)^2

Substituting this into the original equation, we get:

y^2 - 3y - 28 = 0

The equation is now a simple quadratic equation which can be solved by factoring or using the quadratic formula.

Factoring:

(y-7)(y+4) = 0

Therefore, y = 7 or y = -4

Now we substitute back for y to find the values of x:

For y = 7: (x+8)^2 = 7 x+8 = ±√7 x = -8 ±√7

For y = -4: (x+8)^2 = -4 This equation has no real solutions since the square of a real number cannot be negative.

Therefore, the solutions to the original equation are:

x = -8 + √7 x = -8 - √7