Problem 179 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The roots of the equation \(z^{4}+4 z^{2}+16=0\) are (A) \(\pm 1+i \sqrt{3}\) (B) \(\pm 1-i \sqrt{3}\) (C) \(\pm 2+2 \sqrt{3} i\) (D) \(\pm 2-2 \sqrt{3} i\)

Step-by-Step Solution

Answer

The roots are \(\pm 2 + 2\sqrt{3}i\) (option C).

Step 1: Use Substitution to Simplify

Start by letting \(w = z^2\). This substitution transforms the original equation \(z^4 + 4z^2 + 16 = 0\) into a quadratic equation: \(w^2 + 4w + 16 = 0\).

Step 2: Solve the Quadratic Equation

Solve the quadratic equation \(w^2 + 4w + 16 = 0\) using the quadratic formula: \[w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]where \(a = 1\), \(b = 4\), and \(c = 16\). Substitute these values into the formula:

Step 3: Calculate the Discriminant

Compute the discriminant \(b^2 - 4ac\): \(4^2 - 4 \cdot 1 \cdot 16 = 16 - 64 = -48\). The negative discriminant indicates that the roots are complex.

Step 4: Find the Complex Roots

Using the quadratic formula, find the roots: \[w = \frac{-4 \pm \sqrt{-48}}{2} = \frac{-4 \pm i \sqrt{48}}{2}\]Simplify \(\sqrt{48}\) to \(4\sqrt{3}\), giving: \[w = -2 \pm 2i\sqrt{3}\].

Step 5: Convert Back to Original Variable

The solutions \(w = -2 + 2i\sqrt{3}\) and \(w = -2 - 2i\sqrt{3}\) imply \(z^2 = -2 + 2i\sqrt{3}\) and \(z^2 = -2 - 2i\sqrt{3}\). Find \(z\) by taking the square root of these expressions.

Step 6: Calculate Roots of Original Equation

The solutions for \(z^2 = -2 + 2i\sqrt{3}\) are \(z = \pm (\sqrt{-2 + 2i\sqrt{3}})\), and for \(z^2 = -2 - 2i\sqrt{3}\) are \(z = \pm (\sqrt{-2 - 2i\sqrt{3}})\). Evaluating these square roots gives the roots: \(z = \pm (2 + 2i\sqrt{3})\).

Step 7: Match the Roots with the Given Options

Comparing the roots \(z = \pm (2 + 2i\sqrt{3})\) from the solution with the options provided in the exercise leads to the choice corresponding to (C) \(\pm 2 + 2\sqrt{3}i\).

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of degree 2, characterized by the general form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). In the context of our original exercise, we encountered a higher-degree polynomial: \( z^4 + 4z^2 + 16 = 0 \). To simplify this problem, we used a substitution technique by letting \( w = z^2 \). This transformed our high-degree equation into the quadratic equation \( w^2 + 4w + 16 = 0 \).

Here are some key points about quadratic equations:

• They typically have two solutions, known as "roots." These roots can be real or complex.
• Quadratic equations can often be solved using the quadratic formula, factoring, or completing the square method.
• The solutions represent the x-values where the parabola, defined by the quadratic equation, intersects the x-axis.

By simplifying higher-degree equations into quadratic ones, we make the task of solving them more approachable, using well-known methods and techniques.

Complex Numbers

Complex numbers extend the concept of regular numbers by including 'imaginary' components. A complex number is usually expressed in the form \( a + bi \), where \( a \) is the real part, \( b \) is the imaginary part, and \( i \) is the square root of \( -1 \). In our example, the quadratic equation \( w^2 + 4w + 16 = 0 \) had a negative discriminant, indicating complex roots:

• Complex numbers are crucial because they provide solutions to equations that do not have real number solutions. For instance, find the roots of \( z^2 = -2 \cdot (1 - i\sqrt{3}) \).
• The square root of a negative number is defined via the imaginary unit \( i \), as \( \sqrt{-1} = i \).
• In mathematics, complex numbers have practical applications across fields like engineering, physics, and signal processing.
  
  Having a grasp of complex numbers enables us to solve equations with negative discriminants and understand the full range of possible solutions.

Discriminant Analysis

The discriminant is a part of the quadratic formula that helps determine the nature of the roots of a quadratic equation. For an equation \( ax^2 + bx + c = 0 \), the discriminant, \( \Delta \), is given by \( b^2 - 4ac \). The value of the discriminant reveals much about the type of roots we can expect:

• If \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), the equation has exactly one real root (also called a repeated or double root).
• If \( \Delta < 0 \), the equation has two complex conjugate roots.
  

In our original exercise, we computed the discriminant \( b^2 - 4ac \) for the equation \( w^2 + 4w + 16 = 0 \) as \( 16 - 64 = -48 \). The negative value here indicated the presence of complex roots. By recognizing this, we were able to proceed to find those complex solutions accurately.