Without solving the given quadratic equation, determine the number and type of solutions. $9x^2=2-3x$

Verified step by step guidance

1. Rewrite the given quadratic equation in standard form \(ax^2 + bx + c = 0\). Start by moving all terms to one side: \$9x^2 + 3x - 2 = 0$.
2. Identify the coefficients: \(a = 9\), \(b = 3\), and \(c = -2\).
3. Calculate the discriminant using the formula \(\Delta = b^2 - 4ac\).
4. Substitute the values into the discriminant formula: \(\Delta = (3)^2 - 4(9)(-2)\).
5. Analyze the discriminant value: if \(\Delta > 0\), there are two distinct real solutions; if \(\Delta = 0\), there is one real repeated solution; if \(\Delta < 0\), there are two complex solutions.

Key Concepts

Standard Form of a Quadratic Equation

A quadratic equation is typically written in the form ax² + bx + c = 0, where a, b, and c are constants. Converting the given equation into this form is essential for analyzing its properties, such as the number and type of solutions.

Discriminant of a Quadratic Equation

The discriminant, given by Δ = b² - 4ac, determines the nature of the roots of a quadratic equation. If Δ > 0, there are two distinct real solutions; if Δ = 0, there is one real repeated solution; and if Δ < 0, there are two complex conjugate solutions.

Types of Solutions for Quadratic Equations

Quadratic equations can have real or complex solutions. Real solutions occur when the graph of the quadratic intersects the x-axis, while complex solutions occur when it does not. Understanding this helps in interpreting the discriminant without solving the equation.