Final answer:
To solve the equation x⁴ - 3x² + 2 = 0, a substitution of u = x² is used to transform the quartic equation into a quadratic one, u² - 3u + 2 = 0. The quadratic formula is then applied to solve for u, after which x is found by reverting the substitution.
Step-by-step explanation:
To solve the equation x⁴ - 3x² + 2 = 0 using substitution, we look for a substitution that will turn the quartic equation into a quadratic form, which is easier to solve. A natural choice for substitution is u = x², as the equation only involves terms of x to the power of 2 and 4.
By substituting u for x², the original equation becomes u² - 3u + 2 = 0. This is now a quadratic equation in the form of au² + bu + c = 0, where a is 1, b is -3, and c is 2.
To solve for u, we can use the quadratic formula, which is u = (-b ± √(b² - 4ac)) / (2a). After finding the solutions for u, we substitute back x² for u and solve for x to find the original variable's solutions.