%5E2-5(x%5E2-7)%2B6%3D0.jpeg "(x^2-7)^2-5(x^2-7)+6=0")
Solving the Quadratic Equation: (x^2-7)^2 - 5(x^2-7) + 6 = 0
This equation may seem complicated at first glance, but it can be solved using a simple substitution technique. Here's how:
1. Substitution:
Let's simplify the equation by substituting a new variable. Let y = x^2 - 7. Now the equation becomes:
y^2 - 5y + 6 = 0
2. Factoring:
This is a standard quadratic equation. We can factor it easily:
(y - 2)(y - 3) = 0
This gives us two possible solutions for 'y':
- y = 2
- y = 3
3. Back Substitution:
Now, we need to substitute back the original expression for 'y':
- x^2 - 7 = 2
- x^2 - 7 = 3
Solving these equations:
- x^2 = 9
- x^2 = 10
4. Finding the Solutions:
Taking the square root of both sides for each equation:
- x = ± 3
- x = ± √10
Conclusion:
Therefore, the solutions to the equation (x^2-7)^2 - 5(x^2-7) + 6 = 0 are:
- x = 3
- x = -3
- x = √10
- x = -√10