(x^2+x)^2-8(x^2+x)+12=0

2 min read Jun 17, 2024
(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

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