Problem 14 - Algebra 2 and Trigonometry

Question

In \(3-14,\) use the quadratic formula to find the imaginary roots of each equation. $$ 2 x=x^{2}+3 $$

Step-by-Step Solution

Answer

The imaginary roots are \( x = 1 \pm i\sqrt{2} \).

Step 1: Write the equation in standard form

First, we need to rearrange the given equation to the standard form of a quadratic equation, which is \( ax^2 + bx + c = 0 \). The provided equation is \( 2x = x^2 + 3 \). We rearrange this to: \( x^2 - 2x + 3 = 0 \).

Step 2: Identify coefficients

Now that we have the standard form \( x^2 - 2x + 3 = 0 \), we identify the coefficients: \( a = 1 \), \( b = -2 \), and \( c = 3 \). These coefficients will be used in the quadratic formula.

Step 3: Write the quadratic formula

The quadratic formula is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). We'll use this formula to find the roots of the equation.

Step 4: Calculate the discriminant

Calculate the discriminant \( b^2 - 4ac \), which is part of the quadratic formula. Using the coefficients, we have: \((-2)^2 - 4 \cdot 1 \cdot 3 = 4 - 12 = -8\). A negative discriminant indicates the roots are imaginary.

Step 5: Compute the roots using the quadratic formula

Since the discriminant \( -8 \) is negative, the roots are imaginary. Plug the values into the formula: \( x = \frac{-(-2) \pm \sqrt{-8}}{2 \cdot 1} \). Simplifying, we get \( x = \frac{2 \pm \sqrt{-8}}{2} \).

Step 6: Simplify the imaginary roots

To simplify \( \sqrt{-8} \), we express it as \( \sqrt{-1 \times 4 \times 2} = 2i\sqrt{2} \), where \( i \) is the imaginary unit. Substitute this back to get \( x = \frac{2 \pm 2i\sqrt{2}}{2} \), which simplifies to \( x = 1 \pm i\sqrt{2} \).

Key Concepts

Imaginary Roots

In mathematics, when dealing with quadratic equations, there are occasions when the solutions, also known as roots, are not real numbers. This happens when the part of the quadratic formula under the square root (the discriminant) is negative. In such cases, we have what are known as **imaginary roots.** Imaginary roots occur because the square root of a negative number is not a real number but an imaginary number.

• Imaginary numbers are created by multiplying a real number by the imaginary unit, represented as \( i \), where \( i = \sqrt{-1} \).
• For example, in the context of finding roots, when the discriminant is \( -8 \), the square root of \( -8 \) simplifies to \( 2i\sqrt{2} \).

Imaginary roots often come in conjugate pairs, such as \( 1 + i\sqrt{2} \) and \( 1 - i\sqrt{2} \), which arises naturally from the quadratic formula. Thus, these solutions help us fully understand the behavior of quadratic functions beyond the realm of real numbers.

Discriminant

The discriminant is a critical part of the quadratic formula used to determine the nature of the roots of a quadratic equation. It is given by the expression \( b^2 - 4ac \), where \( a \), \( b \), and \( c \) are coefficients from the standard form of the quadratic equation \( ax^2 + bx + c = 0 \). The value of the discriminant provides insight into the types of solutions you can expect:

• If the discriminant is positive, there are two distinct real roots.
• If the discriminant is zero, there is exactly one real root, also known as a repeated or double root.
• If the discriminant is negative, as in our example equation, the roots are imaginary.

The discriminant not only indicates the nature of the roots but also affects their complexity. For the given quadratic equation, we calculated a discriminant of \(-8\), pointing directly to the presence of imaginary roots. Understanding the role of the discriminant thus simplifies the process of predicting the behavior of quadratic equations.

Standard Form

A quadratic equation is most usefulness presented in its **standard form**. This is expressed as \( ax^2 + bx + c = 0 \). Here, each aspect of the equation serves a specific role:

• \( x^2 \) represents the quadratic term.
• \( x \) is the linear term.
• The constant term is \( c \).

Transforming any given quadratic equation into standard form allows you to easily identify the coefficients needed for the quadratic formula. For instance, the exercise starts with \( 2x = x^2 + 3 \), which is rearranged to \( x^2 - 2x + 3 = 0 \).When an equation is expressed in standard form, it simplifies calculation tasks like extracting coefficients, computing the discriminant, and plugging numbers into the quadratic formula. This is a fundamental first step in solving any quadratic equation accurately and efficiently.

Coefficients

In quadratic equations, coefficients are crucial as they directly influence the form and solutions of the equation. They are the numerical factors of the equation, denoted as \( a \), \( b \), and \( c \) in the standard form \( ax^2 + bx + c = 0 \).

• \( a \): The coefficient of the quadratic term \( x^2 \). This value can affect the parabola's width and direction.
• \( b \): The coefficient of the linear term \( x \). It influences the slope and position of the parabola.
• \( c \): The constant term, influencing the y-intercept of the parabola.

In the exercise provided, the equation is \( x^2 - 2x + 3 = 0 \), which corresponds to coefficients \( a = 1 \), \( b = -2 \), and \( c = 3 \). These are plugged into the quadratic formula and discriminant to solve for the roots. Understanding coefficients helps in both manual calculations and in using graphing technology to depict equations. By mastering coefficients, you gain greater control and insight into analyzing quadratic functions.