%5E2-8(x%5E2%2Bx)%2B12%3D0.jpeg "(x^2+x)^2-8(x^2+x)+12=0")
Solving the Quadratic Equation: (x^2 + x)^2 - 8(x^2 + x) + 12 = 0
This equation might look intimidating at first glance, but it can be solved using a simple substitution and factoring.
The Substitution Trick
Notice that the expression (x² + x) appears multiple times in the equation. We can simplify the problem by substituting a new variable, say y, for this expression:
Let y = x² + x
Now, our equation becomes:
y² - 8y + 12 = 0
Factoring the Quadratic Equation
This is a standard quadratic equation, which we can solve by factoring. We need to find two numbers that add up to -8 and multiply to 12. These numbers are -6 and -2:
y² - 6y - 2y + 12 = 0
y(y - 6) - 2(y - 6) = 0
(y - 6)(y - 2) = 0
Therefore, y = 6 or y = 2.
Substituting Back and Solving for x
Now we need to substitute back the original expression for y:
Case 1: y = 6
x² + x = 6
x² + x - 6 = 0
Factoring this quadratic equation, we get:
(x + 3)(x - 2) = 0
Therefore, x = -3 or x = 2.
Case 2: y = 2
x² + x = 2
x² + x - 2 = 0
Factoring this quadratic equation, we get:
(x + 2)(x - 1) = 0
Therefore, x = -2 or x = 1.
Final Solutions
The solutions to the equation (x² + x)² - 8(x² + x) + 12 = 0 are:
- x = -3
- x = 2
- x = -2
- x = 1