In Exercises 71–72, without solving the given quadratic equation, determine the number and type of solutions. 9x^2 = 2-3x

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1. Convert the equation into standard quadratic form ax^2 + bx + c = 0 by moving all terms to one side of the equation. For the equation 9x^2 = 2 - 3x, add 3x and subtract 2 from both sides to get 9x^2 + 3x - 2 = 0.
2. Identify the coefficients from the standard form of the quadratic equation, where a = 9, b = 3, and c = -2.
3. Calculate the discriminant using the formula \(\Delta = b^2 - 4ac\). Substitute the identified coefficients into the formula to find the value of the discriminant.
4. Analyze the value of the discriminant to determine the nature of the solutions: If \(\Delta > 0\), the equation has two distinct real solutions; if \(\Delta = 0\), the equation has exactly one real solution (a repeated root); if \(\Delta < 0\), the equation has two complex conjugate solutions.
5. Based on the calculated discriminant, conclude the number and type of solutions for the quadratic equation without solving it.

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions to these equations can be found using various methods, including factoring, completing the square, or applying the quadratic formula. Understanding the structure of quadratic equations is essential for analyzing their solutions.

Discriminant

The discriminant of a quadratic equation, given by the formula D = b^2 - 4ac, is a key component in determining the nature of the solutions. It indicates whether the solutions are real or complex, and whether they are distinct or repeated. Specifically, if D > 0, there are two distinct real solutions; if D = 0, there is one real solution; and if D < 0, there are two complex solutions.

Standard Form

The standard form of a quadratic equation is crucial for analysis and is typically expressed as ax^2 + bx + c = 0. To analyze the given equation, it may need to be rearranged into this form. This allows for the identification of coefficients a, b, and c, which are necessary for calculating the discriminant and determining the number and type of solutions.