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

Verified step by step guidance

1. Let us start by making a substitution to simplify the equation. Set \( y = x^{1/3} \). This means \( y^2 = (x^{1/3})^2 = x^{2/3} \).
2. Substitute \( y \) and \( y^2 \) into the equation: \( 2y^2 - 5y - 3 = 0 \).
3. Now, we have a quadratic equation in terms of \( y \). We can solve this quadratic equation using the quadratic formula: \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = -5 \), and \( c = -3 \).
4. Calculate the discriminant \( b^2 - 4ac \) to determine the nature of the roots.
5. Once you find the values of \( y \), substitute back \( y = x^{1/3} \) to find the corresponding values of \( x \) by solving \( x = y^3 \).

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 straightforward 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.