In Exercises 75–82, compute the discriminant. Then determine the number and type of solutions for the given equation. $x^2-4x-5=0$

Verified step by step guidance

1. Identify the coefficients in the quadratic equation \(x^2 - 4x - 5 = 0\). Here, \(a = 1\), \(b = -4\), and \(c = -5\).
2. Recall the formula for the discriminant: \(\Delta = b^2 - 4ac\).
3. Substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula: \(\Delta = (-4)^2 - 4(1)(-5)\).
4. Simplify the expression to find the discriminant value (do not calculate the final number here).
5. Use 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 solution; if \(\Delta < 0\), there are two complex solutions.

Key Concepts

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. It represents a parabola when graphed and can have zero, one, or two real solutions depending on its coefficients.

Discriminant

The discriminant is the part of the quadratic formula under the square root, given by b² - 4ac. It determines the nature and number of solutions of a quadratic equation: if positive, two distinct real solutions; if zero, one real repeated solution; if negative, two complex solutions.

Types of Solutions of Quadratic Equations

The solutions of a quadratic equation can be real or complex. Real solutions occur when the discriminant is zero or positive, indicating where the parabola intersects the x-axis. Complex solutions arise when the discriminant is negative, meaning the parabola does not cross the x-axis.