Solve: 2x^2/3 - 5x^1/3 -3 = 0.

Verified step by step guidance

1. Rewrite the equation by substituting a new variable to simplify the exponents. Let \( y = x^{1/3} \), so \( x^{2/3} = y^2 \). The equation becomes \( 2y^2 - 5y - 3 = 0 \).
2. Recognize that the equation \( 2y^2 - 5y - 3 = 0 \) is a quadratic equation. Use the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = -5 \), and \( c = -3 \).
3. Substitute the values of \( a \), \( b \), and \( c \) into the quadratic formula to find \( y \). This gives \( y = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-3)}}{2(2)} \). Simplify the expression under the square root and the rest of the formula.
4. Once you solve for \( y \), remember that \( y = x^{1/3} \). Rewrite the solutions for \( y \) as \( x^{1/3} = y \), and then cube both sides to solve for \( x \). This gives \( x = y^3 \).
5. Verify the solutions by substituting them back into the original equation \( 2x^{2/3} - 5x^{1/3} - 3 = 0 \) to ensure they satisfy the equation. Discard any extraneous solutions if necessary.

Key Concepts

Rational Exponents

Rational exponents are a way to express roots using fractional powers. For example, x^(1/n) represents the n-th root of x. In the given equation, the terms 2x^(2/3) and 5x^(1/3) utilize rational exponents, which can be rewritten in radical form to facilitate solving the equation.

Substitution Method

The substitution method involves replacing a complex expression with a simpler variable to make solving easier. In this case, letting y = x^(1/3) transforms the equation into a quadratic form, allowing for easier application of factoring or the quadratic formula to find solutions.

Quadratic Equations

Quadratic equations are polynomial equations of the form ax^2 + bx + c = 0, where a, b, and c are constants. They can be solved using various methods, including factoring, completing the square, or the quadratic formula. Understanding how to manipulate and solve these equations is crucial for finding the roots of the transformed equation in the problem.