Problem 63 - Algebra 2

Question

Find the value of the discriminant for each quadratic equation. Then describe the number and type of roots for the equation. $$ 2 x^{2}-5 x+6=0 $$

Step-by-Step Solution

Answer

Discriminant is -23, leading to two complex roots.

Step 1: Identify a, b, and c

In the quadratic equation \( ax^2 + bx + c = 0 \), we identify the coefficients: \( a = 2 \), \( b = -5 \), and \( c = 6 \).

Step 2: Use the Discriminant Formula

The discriminant \( D \) of a quadratic equation is given by the formula \( D = b^2 - 4ac \). We will use this formula to calculate the discriminant.

Step 3: Calculate \( b^2 \)

Calculate the square of \( b \): \((-5)^2 = 25\).

Step 4: Calculate \( 4ac \)

Calculate \( 4 \) times \( a \) times \( c \): \( 4 \times 2 \times 6 = 48 \).

Step 5: Evaluate the Discriminant

Substitute the calculated values into the discriminant formula: \( D = 25 - 48 = -23 \).

Step 6: Determine the Nature of the Roots

Since the discriminant \( D = -23 \), which is less than zero, the quadratic equation has two complex roots.

Key Concepts

Nature of Roots

Every quadratic equation may have different types of roots, depending on its discriminant, which is found using the formula \( D = b^2 - 4ac \). You can scatter out simplified pieces of information to understand this concept better:

• If \( D > 0 \), the equation has two distinct real roots.
• If \( D = 0 \), it has exactly one real root, also known as a repeated or double root.
• If \( D < 0 \), the roots are not real - they are complex.

You can think of the discriminant as a kind of "calculator" that predicts how many solutions an equation has and the kind of number those solutions are.
This means by simply glancing at the value of \( D \), you can deduce if you will encounter real numbers or complex numbers as roots.
Learning to quickly assess the nature of roots helps sharpen problem-solving skills, especially when handling numerous quadratic equations.

Complex Roots

Complex roots appear when the discriminant \( D \,<\, 0 \). These roots aren't plain numbers like 2, -5, or \( \frac{1}{3} \).
They come in pairs, and you can imagine them consisting of both real and imaginary parts.

• A complex number looks like this: \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit.
• The imaginary unit \( i \) is defined by \( i^2 = -1 \).

In our case from the example, due to \( D = -23 \), the roots would definitely contain imaginary components which means you won’t spot them on the standard number line as real numbers.
Understanding complex roots gives a deeper insight into the diverse real-world applications, such as physics where complex numbers are used to describe phenomena like wave mechanics.

Quadratic Equation Coefficients

Quadratic equations take this familiar form \( ax^2 + bx + c = 0 \), where \( a, \) \( b, \) and \( c \) stand as the coefficients. Let's break down their roles:

• \( a \) decides the opening direction and width of the parabola.
• \( b \) influences the location of the vertex along the x-axis.
• \( c \) is simply there to set the y-intercept when \( x = 0 \).

Fun Fact: Identifying these coefficients inflates through the exercise of plugging and examining the discriminant \( D = b^2 - 4ac \).
This makes it crucial to know the correct values of \( a, b, \) and \( c \) for successful root analysis.
Lifting the fog on these coefficients equips you to better manipulate equations and predict their behaviors visually without sketching a graph every time!