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:
1. Substitution:
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
2. Solving the Quadratic Equation:
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.
3. Substituting back:
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
4. Final Solutions:
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