Problem 45 - Algebra and Trigonometry with Analytic Geometry

Question

Find the solutions of the equation $$ 4 x^{2}+x+3=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{-1 + i\sqrt{47}}{8} \) and \( x = \frac{-1 - i\sqrt{47}}{8} \).

Step 1: Identify the Quadratic Equation

The given equation is a quadratic equation: \( 4x^2 + x + 3 = 0 \). A quadratic equation is of the form \( ax^2 + bx + c = 0 \) where \( a \), \( b \), and \( c \) are constants.

Step 2: Determine Coefficients

For the equation \( 4x^2 + x + 3 = 0 \), identify the coefficients: \( a = 4 \), \( b = 1 \), and \( c = 3 \). This information will be used in the quadratic formula.

Step 3: Apply the Quadratic Formula

To find the roots of the quadratic equation, use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Step 4: Calculate the Discriminant

The discriminant \( \Delta \) is given by \( b^2 - 4ac \). Substitute the values of \( a \), \( b \), and \( c \) into this formula: \( 1^2 - 4 \times 4 \times 3 = 1 - 48 = -47 \).

Step 5: Evaluate the Discriminant

Since the discriminant \( \Delta = -47 \) is less than zero, the quadratic equation does not have real solutions but instead has two complex conjugate solutions.

Step 6: Find the Complex Solutions

Since the discriminant is negative, use \( x = \frac{-b \pm i\sqrt{-\Delta}}{2a} \) to find the complex solutions. This becomes: \( x = \frac{-1 \pm i\sqrt{47}}{8} \). Thus, the solutions are \( x = \frac{-1 + i\sqrt{47}}{8} \) and \( x = \frac{-1 - i\sqrt{47}}{8} \).

Key Concepts

Complex Solutions

In some cases, quadratic equations may not have real solutions. This happens when the discriminant, a value calculated from the equation's coefficients, is less than zero. If the discriminant is negative, the solutions to the quadratic equation will be complex. Complex numbers go beyond the traditional number line. They include a real part and an imaginary part, the latter involving the imaginary unit, denoted as \(i\), where \(i^2 = -1\).

When the discriminant is negative, the solutions take the form of complex conjugates. These solutions are expressed as:

• \( x = \frac{-b + i\sqrt{-\Delta}}{2a} \)
• \( x = \frac{-b - i\sqrt{-\Delta}}{2a} \)

For the equation \(4x^2 + x + 3 = 0\), with a discriminant of \(-47\), the complex solutions are \( x = \frac{-1 + i\sqrt{47}}{8} \) and \( x = \frac{-1 - i\sqrt{47}}{8} \). This means instead of crossing the x-axis, the parabola associated with the equation lies entirely above or below the axis, indicating the absence of real roots.

Quadratic Formula

The quadratic formula is an essential tool for finding the solutions to any quadratic equation of the form \(ax^2 + bx + c = 0\). This formula provides a straightforward path to uncover the solution, even when the coefficients and constants involved are hard to manage manually. The quadratic formula is given by:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

What makes this formula powerful is that it works universally for all quadratic equations, as long as the appropriate coefficients \(a\), \(b\), and \(c\) are correctly substituted. Depending on the value of the discriminant \(b^2 - 4ac\), the quadratic formula provides real or complex solutions.

In our problem, the quadratic equation \(4x^2 + x + 3 = 0\) makes use of coefficients \(a = 4\), \(b = 1\), and \(c = 3\). Applying these into the quadratic formula, we determine that the solutions are complex, owing to the negative discriminant value calculated from these coefficients.

Discriminant

The discriminant is a component of the quadratic formula that reveals the nature of the solutions of a quadratic equation. It is calculated as \( \Delta = b^2 - 4ac \). The value of the discriminant can take one of three roles:

• If \( \Delta > 0\), there are two distinct real solutions.
• If \( \Delta = 0\), there is exactly one real solution (a repeated root).
• If \( \Delta < 0\), there are no real solutions, but instead two complex conjugate solutions.

Understanding the discriminant's role is crucial as it predicts the type of solutions without actually having to solve the equation. For the quadratic equation \(4x^2 + x + 3 = 0\), calculating the discriminant with \(b = 1\), \(a = 4\), and \(c = 3\) results in \(1 - 48 = -47\). This negative outcome indicates that the equation leads to two complex solutions.