List all numbers that must be excluded from the domain of each rational expression. 3/(2x² + 4x - 9)

Verified step by step guidance

1. Identify the domain of a rational expression: The domain consists of all real numbers except those that make the denominator equal to zero. For this problem, the denominator is 2x^2 + 4x - 9.
2. Set the denominator equal to zero to find the values of x that must be excluded: 2x^2 + 4x - 9 = 0.
3. Simplify the quadratic equation: Divide through by 2 to make the equation simpler, resulting in x^2 + 2x - 4.5 = 0.
4. Solve the quadratic equation using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a, where a = 1, b = 2, and c = -4.5.
5. Calculate the discriminant (b^2 - 4ac) and substitute the values into the quadratic formula to find the two solutions for x. These solutions are the values that must be excluded from the domain.

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. Understanding rational expressions is crucial because they can have restrictions on their domain, specifically where the denominator equals zero, as division by zero is undefined.

Finding the Domain

The domain of a rational expression consists of all the possible input values (x-values) that do not make the denominator zero. To find the domain, one must solve the equation set by the denominator equal to zero and exclude those values from the domain.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler components (factors) that, when multiplied together, give the original polynomial. This is essential for identifying the values that make the denominator zero, as it allows for easier solving of the equation.