Problem 1 - Algebra 2 and Trigonometry

Question

The discriminant of the quadratic equation \(\tan ^{2} \theta+4 \tan \theta+5=0\) is \(-4 .\) Explain why the solution set of this equation is the empty set.

Step-by-Step Solution

Answer

The solution set is empty because the discriminant is negative, indicating no real roots.

Step 1: Understanding the Quadratic Equation

The given equation is a quadratic in terms of \( \tan \theta \), which is written as \( \tan^2 \theta + 4 \tan \theta + 5 = 0 \). The general form of a quadratic equation is \( ax^2 + bx + c = 0 \). Here \( a = 1 \), \( b = 4 \), and \( c = 5 \).

Step 2: Identifying the Discriminant

For any quadratic equation \( ax^2 + bx + c = 0 \), the discriminant \( \Delta \) is given by the formula \( \Delta = b^2 - 4ac \). In our case, substituting the values, we calculate \( \Delta = 4^2 - 4 \times 1 \times 5 = 16 - 20 = -4 \). The discriminant here is \(-4\).

Step 3: Interpreting the Discriminant

The discriminant helps determine the nature of roots of a quadratic equation. When the discriminant is negative (\( \Delta < 0 \)), it means that the quadratic equation has no real roots and only complex or imaginary roots.

Step 4: Conclusion About the Solution Set

Since the equation \( \tan^2 \theta + 4 \tan \theta + 5 = 0 \) has a negative discriminant (\( \Delta = -4 \)), it implies there are no real solutions. Hence, no real number can satisfy this equation, making the real solution set empty.

Key Concepts

Quadratic Equation

A quadratic equation is a foundational concept in algebra. It is characterized by the general form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). The variable is most often denoted as \( x \), but in many contexts, such as trigonometry, it might be \( \tan \theta \) like in our exercise.

• In our case, the equation is \( \tan^2 \theta + 4 \tan \theta + 5 = 0 \), similar to the standard form but focused on \( \tan \theta \) instead of \( x \).
• The coefficients for our specific equation are \( a = 1 \), \( b = 4 \), and \( c = 5 \).

Quadratic equations can have different types of roots (solutions), and we can analyze them using several methods, one of which involves the discriminant.

Nature of Roots

The nature of the roots of a quadratic equation gives us crucial information about its solutions. This is determined by the discriminant, \( \Delta = b^2 - 4ac \). Depending on the value of \( \Delta \), we can predict the nature of the roots:

• \( \Delta > 0 \): Two distinct real roots exist.
• \( \Delta = 0 \): Exactly one real root, also called a repeated or double root.
• \( \Delta < 0 \): No real roots, but two complex roots.

In our equation, \( \Delta = -4 \), which means the roots are complex. Therefore, there are no real solutions, hence the real solution set is empty for this equation.

Complex Roots

When quadratic equations have a negative discriminant, they produce complex roots. Complex roots occur in conjugate pairs of the form \( a + bi \) and \( a - bi \), where \( i \) is the imaginary unit satisfying \( i^2 = -1 \). Almost every time a quadratic has complex roots, the real parts are the same, while the imaginary parts are equal and opposite.

This indicates that solutions aren’t visible on the real number line. Instead, they exist within the complex plane. Because these roots aren't real, they're essential in fields like engineering and physics, but they don't solve equations where we're seeking real-world measurements or quantities.

Tan Function

The tangent function, denoted as \( \tan \theta \), is a basic trigonometric function that relates to angles in right-angled triangles or points on the unit circle. It is essential in calculus, geometry, and trigonometry, serving as the ratio of the sine and cosine functions, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

• \( \tan \theta \) is periodic with a period of \( \pi \), meaning it repeats every \( \pi \) radians or \( 180^\circ \).
• The graph of \( \tan \theta \) experiences vertical asymptotes every \( \frac{\pi}{2} + k\pi \), where \( k \) is an integer.
• Used here as the variable in our quadratic equation, \( \tan \theta \) leads to real-world applications, such as calculating slopes and tilt in engineering.

Although in our exercise \( \tan \theta \) leads to complex roots, understanding it is vital in analysing angles and lengths within various scientific contexts.