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 given equation in standard quadratic form \(ax^2 + bx + c = 0\). Start by moving all terms to one side: \$8x^2 + 2x + 6 = 0$.
2. Identify the coefficients \(a\), \(b\), and \(c\) from the standard form. Here, \(a = 8\), \(b = 2\), and \(c = 6\).
3. Recall the formula for the discriminant: \(\Delta = b^2 - 4ac\). Substitute the values of \(a\), \(b\), and \(c\) into this formula.
4. Calculate the discriminant expression: \(\Delta = (2)^2 - 4 \times 8 \times 6\) (do not simplify the final value).
5. Use the value of the discriminant to determine the number and type of solutions: 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

Quadratic Equation Standard Form

A quadratic equation is typically written as ax² + bx + c = 0, where a, b, and c are constants. To analyze the equation, it must first be rearranged into this standard form by moving all terms to one side. This form is essential for applying formulas like the discriminant.

Discriminant of a Quadratic Equation

The discriminant is given by the formula Δ = b² - 4ac and helps determine the nature of the roots of a quadratic equation. A positive discriminant indicates two distinct real solutions, zero means one real repeated solution, and a negative discriminant implies two complex conjugate solutions.

Interpreting the Number and Type of Solutions

Using the value of the discriminant, one can classify the solutions of the quadratic equation. This classification informs whether the solutions are real or complex and whether they are distinct or repeated, which is crucial for understanding the behavior of the quadratic function.