Solve each equation. (2x-1)^2/3+2(2x-1)^1/3-3=0

검증된 단계별 안내

Start by making a substitution to simplify the equation. Let \(y = (2x - 1)^{1/3}\), which means \(y^2 = (2x - 1)^{2/3}\).

Rewrite the original equation in terms of \(y\): \(y^2 + 2y - 3 = 0\).

Recognize that this is a quadratic equation in \(y\). Use factoring or the quadratic formula to solve for \(y\).

After finding the values of \(y\), substitute back to get \((2x - 1)^{1/3} = y\). Then, cube both sides to solve for \(2x - 1\).

Finally, solve the resulting linear equations for \(x\) by isolating \(x\) on one side.

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주요 개념

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

Rational Exponents

Rational exponents represent roots and powers simultaneously, where an exponent like m/n means the nth root raised to the mth power. For example, x^(2/3) means the cube root of x squared. Understanding how to manipulate these exponents is essential for simplifying and solving equations involving fractional powers.

Substitution Method

The substitution method involves replacing a complex expression with a single variable to simplify the equation. In this problem, letting y = (2x - 1)^(1/3) transforms the equation into a quadratic form, making it easier to solve. After solving for y, substitute back to find x.

Solving Quadratic Equations

Quadratic equations are polynomial equations of degree two and can be solved by factoring, completing the square, or using the quadratic formula. Once the substitution reduces the original equation to a quadratic in y, these methods help find the values of y, which then lead to the solutions for x.

Solve each equation. (2x-1)2/3+2(2x-1)1/3-3=0