Problem 12 - Algebra 2 and Trigonometry

Question

In \(3-14,\) use the quadratic formula to find the imaginary roots of each equation. $$ x^{2}+5=4 x $$

Step-by-Step Solution

Answer

The imaginary roots are \( x = 2 + i \) and \( x = 2 - i \).

Step 1: Write the Equation in Standard Form

First, we need to rewrite the quadratic equation given in the standard form, which is \( ax^2 + bx + c = 0 \). The original equation is \( x^2 + 5 = 4x \). Subtract \( 4x \) from both sides to get \( x^2 - 4x + 5 = 0 \).

Step 2: Identify the Coefficients

Now that we have the quadratic equation in the form \( ax^2 + bx + c = 0 \), identify the coefficients: \( a = 1 \), \( b = -4 \), and \( c = 5 \).

Step 3: Use the Quadratic Formula

The quadratic formula is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute the values of \( a \), \( b \), and \( c \) into the formula: \( x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 5}}{2 \cdot 1} \).

Step 4: Simplify the Expression

Calculate inside the square root: \( (-4)^2 = 16 \) and \( 4 \cdot 1 \cdot 5 = 20 \). This gives us \( x = \frac{4 \pm \sqrt{16 - 20}}{2} \), or \( x = \frac{4 \pm \sqrt{-4}}{2} \).

Step 5: Solve for Imaginary Roots

The square root of \(-4\) is an imaginary number: \( \sqrt{-4} = 2i \). So, our equation becomes \( x = \frac{4 \pm 2i}{2} \). Simplifying this gives us the solutions: \( x = 2 + i \) and \( x = 2 - i \).

Key Concepts

Imaginary Roots

When you encounter a quadratic equation and need to find its roots, you might discover that they can be imaginary. Imaginary roots occur when the discriminant of a quadratic equation is negative. In the quadratic formula, the discriminant is the part under the square root: \(b^2 - 4ac\).
In our example, the discriminant \(16 - 20\) equals \(-4\). This negative value means that the square root involves an imaginary number. Imaginary numbers use "\(i\)," where \(i\) is the square root of \(-1\).
Some key points to remember about imaginary roots:

• Imaginary numbers are paired with real numbers when forming solutions. These are also called complex numbers.
• Imaginary roots often occur as conjugates, which means if \(x = a + bi\) is a root, \(x = a - bi\) is the other root.
• They help represent roots that cannot be placed on the traditional number line.

Understanding imaginary roots is crucial as they appear often in problems involving quadratic equations.

Standard Form Quadratic Equation

A quadratic equation is any equation that can be represented in the form \(ax^2 + bx + c = 0\). This is called the standard form. It's a structured way to write a quadratic equation which is essential when applying the quadratic formula to find roots.
The standard form makes it easy to identify:

• "\(a\)" as the coefficient of \(x^2\), determining the parabola's opening direction and width.
• "\(b\)" as the coefficient of \(x\), influencing the symmetry of the parabola.
• "\(c\)" as the constant term, impacting the y-intercept.

Converting a given equation into this form often involves simple algebraic manipulation. For instance, from \(x^2 + 5 = 4x\) to \(x^2 - 4x + 5 = 0\). Recognizing it in this standard form is the first step in applying the quadratic formula. It ensures precise computation when determining the solution types: real or complex roots.

Complex Numbers

Complex numbers are extensions of the real number system, incorporating imaginary numbers. They take the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) represents the square root of \(-1\). In quadratic equations, complex numbers frequently appear as solutions when the discriminant is negative, as seen in our example.
Important attributes of complex numbers include:

• The real part is the "\(a\)" term, and the imaginary part is the "\(b\)" term.
• The "\(i\)" stands for an imaginary unit, which is fundamental in describing numbers not found on the traditional number line.
• Complex conjugates are used in solutions like \(x = 2 + i\) and \(x = 2 - i\), ensuring the quadratic equation's solutions are comprehensive.

Having a grasp on complex numbers not only aids in solving quadratic equations but also prepares you to tackle a broad range of advanced mathematical concepts.