%5E2%2Bx%5E2-8%3D20.jpeg "(x^2-8)^2+x^2-8=20")
Solving the Equation: (x^2-8)^2+x^2-8=20
This equation might look intimidating at first glance, but it can be solved using a simple substitution. Here's how:
1. Substitution
Let's make the equation easier to work with. We can substitute y = x^2 - 8. This transforms the equation into:
y^2 + y = 20
2. Rearranging and Factoring
Now we have a quadratic equation:
y^2 + y - 20 = 0
This can be factored into:
(y + 5)(y - 4) = 0
3. Solving for y
This gives us two possible solutions for y:
- y = -5
- y = 4
4. Back-Substituting to Find x
Now we need to substitute back our original expression for y:
- x^2 - 8 = -5
- x^2 - 8 = 4
Solving for x in each case:
- x^2 = 3 => x = ±√3
- x^2 = 12 => x = ±2√3
5. The Solutions
Therefore, the solutions to the equation (x^2-8)^2+x^2-8=20 are:
- x = √3
- x = -√3
- x = 2√3
- x = -2√3