(x^2+x)^2-5(x^2+x)+6=0

Jasonbradley

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Jun 17, 2024

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Solving the Quadratic Equation: (x^2 + x)^2 - 5(x^2 + x) + 6 = 0

This equation may look intimidating at first, 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. We'll replace (x² + x) with a new variable, let's say 'y':

• Let y = (x² + x)

Now, the equation becomes:

y² - 5y + 6 = 0

2. Factoring the Quadratic

This is a standard quadratic equation, and we can factor it:

(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':

• For y = 2:

  • x² + x = 2
  • x² + x - 2 = 0
  • (x + 2)(x - 1) = 0
  • This gives us x = -2 and x = 1

• For y = 3:

  • x² + x = 3
  • x² + x - 3 = 0
  • This equation doesn't factor easily, so we'll use the quadratic formula:
    • x = (-b ± √(b² - 4ac)) / 2a
    • Where a = 1, b = 1, and c = -3
    • This gives us x = (-1 ± √13) / 2

4. Solutions

Therefore, the solutions for the original equation (x² + x)² - 5(x² + x) + 6 = 0 are:

• x = -2
• x = 1
• x = (-1 + √13) / 2
• x = (-1 - √13) / 2