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(x2−7)2+2x2−14=0

2 min read Jun 17, 2024
(x2−7)2+2x2−14=0

Solving the Equation (x² - 7)² + 2x² - 14 = 0

This equation looks complex at first glance, but we can solve it by employing a few algebraic tricks. Here's how:

1. Simplifying the Equation

  • Recognize the pattern: Notice that the expression (x² - 7)² is a perfect square. We can rewrite the equation as: (x² - 7)² + 2(x² - 7) = 0

  • Substitution: Let's make a substitution to simplify the equation further. Let y = (x² - 7). Now the equation becomes: y² + 2y = 0

2. Solving the Quadratic Equation

  • Factoring: We can factor out a 'y' from the equation: y(y + 2) = 0

  • Zero Product Property: For the product of two factors to be zero, at least one of them must be zero. Therefore, we have two possible solutions:

    • y = 0
    • y + 2 = 0

3. Substituting Back and Solving for x

  • Substitute y back: Now we need to substitute back (x² - 7) for y in both equations:

    • x² - 7 = 0
    • x² - 7 + 2 = 0

  • Solve for x:

    • Equation 1: x² = 7 => x = ±√7
    • Equation 2: x² = 5 => x = ±√5

4. Final Solution

The solutions to the equation (x² - 7)² + 2x² - 14 = 0 are:

x = √7, x = -√7, x = √5, x = -√5

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