(x^2-8)^2-5(x^2-8)-14=0

3 min read Jun 17, 2024

Solving the Quadratic Equation: (x^2 - 8)^2 - 5(x^2 - 8) - 14 = 0

This equation might look intimidating at first, but it's actually a disguised quadratic equation. Let's break down the steps to solve it:

The key is to recognize the repeated term: (x^2 - 8). We can substitute this expression with a simpler variable to simplify the equation.

Let y = (x^2 - 8).

This transforms the equation into:

y^2 - 5y - 14 = 0

Now we have a standard quadratic equation in terms of 'y'. We can solve this using various methods, like:

Factoring: Find two numbers that add up to -5 and multiply to -14. These numbers are -7 and 2. Therefore, we can factor the equation as: (y - 7)(y + 2) = 0 This gives us two possible solutions: y = 7 or y = -2.
Quadratic Formula: If factoring is difficult, we can use the quadratic formula: y = (-b ± √(b^2 - 4ac)) / 2a Where a = 1, b = -5, and c = -14. Plugging these values into the formula will also yield the same solutions: y = 7 and y = -2.

Now we need to substitute back the original expression for 'y'.

For y = 7: (x^2 - 8) = 7 x^2 = 15 x = ±√15

For y = -2: (x^2 - 8) = -2 x^2 = 6 x = ±√6

Therefore, the solutions to the original equation (x^2 - 8)^2 - 5(x^2 - 8) - 14 = 0 are:

x = √15, x = -√15, x = √6, x = -√6