Problem 32 - Algebra 2

Question

Evaluate the discriminant of each equation. Tell how many solutions each equation has and whether the solutions are real or imaginary. $$ x^{2}-4 x-5=0 $$

Step-by-Step Solution

Answer

The discriminant is 36, which is greater than 0, indicating that the equation has two distinct real solutions.

Step 1: Identify the coefficients

In the quadratic equation of the form ax^2 + bx + c = 0, identify the values of a, b, and c. Here, a = 1, b = -4, and c = -5.

Step 2: Calculate the discriminant

The discriminant of a quadratic equation is given by the formula D = b^2 - 4ac. Substitute a, b, and c into this formula to calculate the discriminant: D = (-4)^2 - 4(1)(-5).

Step 3: Evaluate the discriminant

Perform the calculations: D = 16 + 20 = 36.

Step 4: Determine the number and type of solutions

Since the discriminant is positive (D > 0), there are two distinct real solutions.

Key Concepts

Quadratic Equation

When we talk about quadratic equations, we refer to polynomials of the second degree, which means they have the highest exponent of 2 for the variable. The general form of the quadratic equation is \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients, and \( a \) is not zero. This is because if \( a \) is zero, the equation would be linear, not quadratic. Quadratic equations are known for their characteristic parabolic graph that opens upwards if \( a \) is positive and downwards if \( a \) is negative.

These equations can be solved by various methods, including factoring, completing the square, using the quadratic formula, or graphing. Each method provides a way to find the values of the variable \( x \) that make the equation true. These values are known as the 'roots' or 'solutions' of the equation. Understanding how to identify and work with the different parts of a quadratic equation is essential for solving them and interpreting their solutions.

Real and Imaginary Solutions

Quadratic equations can have different types of solutions—real or imaginary. If a solution can be placed on the number line, it is considered real. Imaginary solutions, on the other hand, involve the square root of a negative number, which does not have a place on the traditional number line, and therefore requires the imaginary unit \( i \), where \( i^2 = -1 \).

The type of solutions an equation has is determined by the value of the discriminant. If the discriminant is positive, as in the given example, the quadratic equation has two distinct real solutions. These solutions could be rational numbers, like \( 2 \) or \( 3 \) or irrational numbers like \( \sqrt{2} \) or \( \pi \). If the discriminant is zero, the equation has exactly one real solution, where both roots are the same (also known as a repeated root). In contrast, if the discriminant is negative, the quadratic equation has two distinct imaginary solutions, which means they cannot be represented as points on a real number line but instead involve imaginary numbers.

Calculating Discriminant

To understand the nature of the roots of a quadratic equation without actually solving it, we calculate the discriminant. The discriminant is the part of the quadratic formula found under the square root sign and is denoted by \( D \) or sometimes \( \Delta \). It is calculated using the coefficients of the quadratic equation through the formula \( D = b^2 - 4ac \).

In the example \( x^2 - 4x - 5 = 0 \) we identified that \( a = 1 \) , \( b = -4 \) , and \( c = -5 \) . Substituting these into the discriminant formula, \( D \) equals \( (-4)^2 - 4(1)(-5) \) which simplifies to \( 16 + 20 \) resulting in a discriminant of \( 36 \) .

Interpreting the Discriminant

Since the computed discriminant is positive (\( D > 0 \)), we confirm that there are two distinct real roots for the quadratic equation. This means that the parabola represented by the equation will intersect the x-axis at two points. These points of intersection are the real solutions the question refers to. Calculating the discriminant provides a quick and efficient way to predict the number and types of solutions for any quadratic equation without needing to solve it completely.