Problem 75 - Algebra and Trigonometry

Question

Compute the discriminant. Then determine the number and type of solutions for the given equation. $$ x^{2}-4 x-5=0 $$

Step-by-Step Solution

Answer

The discriminant of the given equation is 36. Therefore, the equation has two distinct real solutions.

Step 1: Identify the coefficients

The given equation is \(x^{2} - 4x - 5 = 0\). In standard form, the coefficients are \(a = 1\), \(b = -4\), and \(c = -5\).

Step 2: Compute the discriminant

The discriminant is given by the formula \( \Delta = b^{2} - 4ac \). Substituting \(a = 1\), \(b = -4\), and \(c = -5\) into the formula we get \( \Delta = (-4)^{2} - 4*1*(-5) = 16 + 20 = 36.\)

Step 3: Determine the number and type of solutions

Since the result is positive, the equation has 2 distinct real solutions.

Key Concepts

Quadratic Equation Solutions

Quadratic equations are fundamental in algebra and they often take the form of \( ax^2 + bx + c = 0 \). Finding the solutions, also known as roots, involves determining the values of \(x\) that make the equation true.

To solve such equations, we can use several methods, including factoring, completing the square, applying the quadratic formula, and graphing. The quadratic formula, which is \(x = \frac{{-b \pm \sqrt{{b^2-4ac}}}}{{2a}}\), can find the roots regardless of whether they are real or complex numbers.

The nature of the solutions is directly associated with the discriminant \(\Delta = b^2 - 4ac\). If \(\Delta > 0\), there are two distinct real solutions. If \(\Delta = 0\), there is one real solution, often called a repeated or double root. For \(\Delta < 0\), the equation has two complex solutions. These properties enable us to anticipate the types and the number of solutions without necessarily computing them expressly.

Computing the Discriminant

The discriminant in algebra is a quantity often denoted as \(\Delta\), calculated from the coefficients of a quadratic equation \(ax^2 + bx + c = 0\).

The formula for computing the discriminant is \(\Delta = b^2 - 4ac\). This value determines the nature of the roots of the quadratic equation without solving it explicitly. When \(\Delta\) is positive, we expect two real and distinct roots. A zero discriminant indicates a single real root, while a negative discriminant implies that the roots are complex and conjugate to each other.

To calculate the discriminant for the equation \(x^2 - 4x - 5 = 0\), identify the coefficients \(a = 1\), \(b = -4\), and \(c = -5\). By substituting these into the discriminant formula, we obtain \(\Delta = (-4)^2 - 4 \times 1 \times (-5) = 36\), which is positive, indicating two real solutions.

Types of Roots in Algebra

In algebra, roots of an equation are the solutions that satisfy the equation. Specifically for quadratic equations, the roots can be real or complex, and their nature can be inferred from the discriminant \(\Delta\).

• If \(\Delta > 0\), the quadratic equation has two distinct real roots, implying the parabola crosses the x-axis at two points.
• If \(\Delta = 0\), the equation has exactly one real root, known as a repeated root, meaning the vertex of the parabola touches the x-axis.
• If \(\Delta < 0\), there are no real roots; instead, there are two complex conjugate roots, indicating the parabola does not intersect with the x-axis at any point.

In the given equation \(x^2 - 4x - 5 = 0\), with the discriminant computed as 36, we conclude that there are two distinct real roots. This is because a positive discriminant correlates to the parabola intersecting the x-axis at two separate points, which is also confirmed graphically or by solving the equation.