Problem 36 - Beginning and Intermediate Algebra

Question

Use the quadratic formula to solve each equation. (All solutions for these equations are non real complex numbers.) $$ x^{2}-5 x+20=0 $$

Step-by-Step Solution

Answer

The solutions are \[x = \frac{5}{2} + \frac{\sqrt{55}}{2} i\] and \[x = \frac{5}{2} - \frac{\sqrt{55}}{2} i\].

Step 1: Identify coefficients

In the quadratic equation \(ax^{2} + bx + c = 0\), identify coefficients \(a\), \(b\), and \(c\). For \(x^{2} - 5x + 20 = 0\), \(a = 1\), \(b = -5\), and \(c = 20\).

Step 2: Write the quadratic formula

The quadratic formula to find the roots of a quadratic equation is \[x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\].

Step 3: Substitute the coefficients into the formula

Substitute \(a = 1\), \(b = -5\), and \(c = 20\) into the quadratic formula: \[x = \frac{-(-5) \pm \sqrt{(-5)^{2} - 4(1)(20)}}{2(1)}\].

Step 4: Simplify inside the square root

Calculate the discriminant, \(b^{2} - 4ac\): \[(-5)^{2} - 4(1)(20) = 25 - 80 = -55\]. Note that the discriminant is negative, indicating complex solutions.

Step 5: Solve for the roots

Continuing with the quadratic formula, we get: \[x = \frac{5 \pm \sqrt{-55}}{2}\]. Express the square root of a negative number using imaginary unit \(i\): \[ \sqrt{-55} = \sqrt{55} i \]. Therefore, \[x = \frac{5 \pm \sqrt{55} i}{2}\].

Step 6: Write the final complex solutions

Separate the solutions into two parts: \[x = \frac{5}{2} + \frac{\sqrt{55}}{2} i\] and \[x = \frac{5}{2} - \frac{\sqrt{55}}{2} i\].

Key Concepts

Complex Numbers

A complex number is a number that can be expressed in the form of \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. Complex numbers extend our number system beyond just real numbers and allow us to solve equations that do not have real solutions. They are essential in many fields, including engineering and physics.
In the given problem, when you end up with a negative number inside the square root, it means the solutions are complex numbers. For example, the equation results in \( \sqrt{-55} \). Since you cannot take the square root of a negative number within the real number system, you use complex numbers to find the solution.

Discriminant

The discriminant is a part of the quadratic formula that indicates the nature of the solutions for a quadratic equation. It is given by \(b^2 - 4ac\). The value of the discriminant helps you understand whether the solutions are real or complex, and if they are real, whether they are distinct or repeated.
In this problem, the discriminant is calculated as follows:

• First, square the value of \(b\): \((-5)^2 = 25\)
• Next, calculate \(4ac\): \(4(1)(20) = 80\)
• Finally, subtract \(4ac\) from \(b^2\): \(25 - 80 = -55\)

The negative discriminant of \(-55\) indicates that the quadratic equation has two complex solutions.

Imaginary Unit

The imaginary unit \(i\) is defined as \(\sqrt{-1}\). It helps us handle the square root of negative numbers and is an essential part of complex numbers.
When the discriminant of a quadratic equation is negative, we use the imaginary unit to express the solutions. In the current problem, \(\sqrt{-55}\) can be expressed using the imaginary unit:

• Rewrite \(\sqrt{-55}\) as \(\sqrt{55} \times \sqrt{-1}\)
• Since \(\sqrt{-1} = i\), this becomes \(\sqrt{55} i\)

The final complex solutions are written as \( \frac{5}{2} + \frac{\sqrt{55}}{2} i \) and \( \frac{5}{2} - \frac{\sqrt{55}}{2} i \). This shows the roots in the form of complex numbers.