(x-3)^4-3(x-3)^2-10=0

2 min read Jun 17, 2024

Solving the Equation (x-3)^4 - 3(x-3)^2 - 10 = 0

This equation looks complex at first glance, but it can be solved using a simple substitution technique. Here's how:

Let's substitute y = (x-3)^2. This simplifies the equation:

y^2 - 3y - 10 = 0

Now we have a standard quadratic equation. We can solve it using the quadratic formula:

y = [-b ± √(b^2 - 4ac)] / 2a

Where a = 1, b = -3, and c = -10

Solving for y, we get:

y = [3 ± √((-3)^2 - 4 * 1 * -10)] / 2 * 1 y = [3 ± √(49)] / 2 y = [3 ± 7] / 2

Therefore, y = 5 or y = -2

Now we need to substitute back the value of y to find x:

Case 1: y = 5

(x-3)^2 = 5 x - 3 = ±√5 x = 3 ± √5

Case 2: y = -2

(x-3)^2 = -2 This equation has no real solutions because the square of any real number cannot be negative.

The solutions to the original equation are:

x = 3 + √5 x = 3 - √5