Solve each equation in Exercises 92–93 by making an appropriate substitution. x^4 - 5x^2 + 4 = 0

검증된 단계별 안내

Recognize that the equation is a quadratic in form, where the variable is x^2 instead of x. To simplify, make the substitution u = x^2. This transforms the equation into u^2 - 5u + 4 = 0.

Now solve the quadratic equation u^2 - 5u + 4 = 0 using factoring, the quadratic formula, or completing the square. For factoring, look for two numbers that multiply to 4 and add to -5.

Once the quadratic equation is factored or solved, find the values of u. These values represent x^2 because of the substitution u = x^2.

Revert the substitution by solving x^2 = u for each value of u. This involves taking the square root of both sides, remembering to include both the positive and negative roots.

Write the final solutions for x, which will include all possible values from the square root calculations. Ensure all solutions are listed clearly.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Substitution Method

The substitution method is a technique used to simplify complex equations by replacing a variable or expression with a new variable. In this case, substituting x^2 with a new variable, such as y, can transform the quartic equation into a quadratic one, making it easier to solve.

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants. The solutions to quadratic equations can be found using factoring, completing the square, or the quadratic formula. Understanding how to manipulate and solve these equations is crucial for solving higher-degree polynomials.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors. This is particularly useful for solving equations, as it allows us to set each factor equal to zero to find the roots. In the context of the given equation, recognizing patterns or using techniques like grouping can help simplify the problem.