(x-7)^4-(x-7)^2-30=0

2 min read Jun 17, 2024

Solving the Equation: (x-7)^4 - (x-7)^2 - 30 = 0

This equation might look intimidating at first, but with a clever substitution, it becomes much easier to solve. Let's break down the steps:

Let **y = (x-7)**². This substitution allows us to rewrite the equation as a quadratic equation:

y² - y - 30 = 0

Now we have a standard quadratic equation that we can solve using the quadratic formula:

y = (-b ± √(b² - 4ac)) / 2a

Where a = 1, b = -1, and c = -30.

Plugging in these values, we get:

y = (1 ± √(1 + 120)) / 2

y = (1 ± √121) / 2

y = (1 ± 11) / 2

This gives us two possible solutions for y:

Now that we have values for y, we need to substitute back to find the solutions for x.

For y₁ = 6:

(x-7)² = 6

Taking the square root of both sides:

x-7 = ±√6

x = 7 ± √6

For y₂ = -5:

(x-7)² = -5

This equation has no real solutions since the square of a real number cannot be negative.

Therefore, the solutions to the original equation (x-7)⁴ - (x-7)² - 30 = 0 are: