Problem 24 - Algebra 2

Question

Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula. \(x^{2}-x+6=0\)

Step-by-Step Solution

Answer

The discriminant is -23, indicating two complex roots: \(\frac{1 + i\sqrt{23}}{2}\) and \(\frac{1 - i\sqrt{23}}{2}\).

Step 1: Identify Coefficients

Identify the coefficients of the quadratic equation. In the equation \(x^{2}-x+6=0\), the coefficients are \(a=1\), \(b=-1\), and \(c=6\).

Step 2: Calculate the Discriminant

Use the formula for the discriminant, \(D = b^2 - 4ac\). Substitute the coefficients: \(D = (-1)^2 - 4 \times 1 \times 6\). Calculate \(D = 1 - 24 = -23\).

Step 3: Determine the Nature of the Roots

Since the discriminant \(D = -23\) is less than zero, this indicates that the equation has no real roots. It has two complex conjugate roots.

Step 4: Apply the Quadratic Formula

Use the quadratic formula \(x = \frac{-b \pm \sqrt{D}}{2a}\) to find the roots. Substitute the values: \(x = \frac{-(-1) \pm \sqrt{-23}}{2 \times 1}\). Simplify to get \(x = \frac{1 \pm i\sqrt{23}}{2}\).

Step 5: Write the Exact Solutions

The exact solutions of the quadratic equation are \(x = \frac{1 + i\sqrt{23}}{2}\) and \(x = \frac{1 - i\sqrt{23}}{2}\).

Key Concepts

Understanding the Discriminant

The discriminant is a key component when working with quadratic equations, as it can tell us important details about the nature of the roots. You can find the discriminant by using the formula \( D = b^2 - 4ac \), which uses the coefficients \( a \), \( b \), and \( c \) from the quadratic equation in the form \( ax^2 + bx + c = 0 \). In this particular problem, for the equation \( x^2 - x + 6 = 0 \), we identified \( a = 1 \), \( b = -1 \), and \( c = 6 \). By substituting these values into the formula, the discriminant calculation is \( D = (-1)^2 - 4 \times 1 \times 6 \). This simplifies to \( D = 1 - 24 \), giving us \( D = -23 \).
Since the discriminant is negative, this indicates the equation will not have any real roots. Instead, with a negative discriminant, the roots will be complex numbers. Understanding the value of the discriminant is crucial as it directly influences the type and number of roots of the quadratic equation.

Exploring the Nature of Roots

The nature of the roots in a quadratic equation hands us insight about the solutions of the equation. By looking at the value of the discriminant, we can easily determine what type of roots the equation has. There are three possibilities to consider:

• If the discriminant \( D \) is greater than zero, the equation will have two distinct real roots.
• If \( D \) is equal to zero, it will have exactly one real root, meaning the roots are real and repeated.
• If \( D \) is less than zero, as in our example \( D = -23 \), the equation results in complex conjugate roots.

In this exercise, since the discriminant \( D \) is less than zero, it reveals that the quadratic equation has complex roots. Specifically, these roots are a pair of complex conjugates, which can be expressed in the form \( rac{1 + i ext{something}}{2} \) and \( rac{1 - i ext{something}}{2} \) where \( i \) represents the imaginary unit. Being able to recognize the nature of the roots by using the discriminant is an effective way to predict possible solution types.

Using the Quadratic Formula for Solutions

The quadratic formula is a universal tool for finding the solutions of any quadratic equation of the form \( ax^2 + bx + c = 0 \). It is written as \( x = \frac{-b \pm \sqrt{D}}{2a} \), where \( D \) is the discriminant. This formula allows us to solve the equation and find exact roots regardless of their nature. In our exercise, we substitute the coefficients from the equation \( x^2 - x + 6 = 0 \), revealing values \( a = 1 \), \( b = -1 \), and \( c = 6 \). We also calculated the discriminant \( D = -23 \).
By using these in the quadratic formula, it translates to \( x = \frac{-(-1) \pm \sqrt{-23}}{2 \times 1} \), further simplifying to \( x = \frac{1 \pm i\sqrt{23}}{2} \). This equation shows that the roots are complex numbers of the form \( x = \frac{1 + i\sqrt{23}}{2} \) and \( x = \frac{1 - i\sqrt{23}}{2} \). Thus, the quadratic formula not only provides us with a clean method to finding solutions but also illustrates how mathematical tools handle different kinds of roots with elegance.