Evaluate the discriminant for each equation. Then use it to determine the number and type of solutions. 8x² = -2x -6

Verified step by step guidance

1. Rewrite the equation in standard quadratic form: \(8x^2 + 2x + 6 = 0\).
2. Identify the coefficients: \(a = 8\), \(b = 2\), \(c = 6\).
3. Use the discriminant formula: \(b^2 - 4ac\).
4. Substitute the coefficients into the discriminant formula: \(2^2 - 4(8)(6)\).
5. Evaluate the discriminant to determine the number and type of solutions: If the discriminant is positive, there are two real solutions; if zero, one real solution; if negative, two complex solutions.

Key Concepts

Discriminant

The discriminant is a key component of the quadratic formula, given by the expression b² - 4ac, where a, b, and c are coefficients from a quadratic equation in the standard form ax² + bx + c = 0. It helps determine the nature of the roots of the equation: if the discriminant is positive, there are two distinct real solutions; if it is zero, there is exactly one real solution; and if it is negative, there are two complex solutions.

Quadratic Equation

A quadratic equation is a polynomial equation of degree two, typically expressed in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions to a quadratic equation can be found using various methods, including factoring, completing the square, or applying the quadratic formula, which incorporates the discriminant to assess the nature of the solutions.

Standard Form

The standard form of a quadratic equation is essential for applying the discriminant effectively. It is represented as ax² + bx + c = 0, where the coefficients a, b, and c are identified. In the given equation, 8x² = -2x - 6, it must first be rearranged into standard form by moving all terms to one side, allowing for the identification of a, b, and c to evaluate the discriminant.