Problem 45 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Equation

The given equation is a quadratic equation of the form \(ax^2 + bx + c = 0\), where \(a = 4\), \(b = 1\), and \(c = 3\).

Step 2: Check for Real Solutions Using the Discriminant

The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(D = b^2 - 4ac\). For the equation \(4x^2 + x + 3 = 0\), compute \(D = 1^2 - 4 \cdot 4 \cdot 3 = 1 - 48 = -47\).

Step 3: Analyzing the Discriminant

The discriminant \(D = -47\) is less than zero, indicating that the quadratic equation has no real roots. Solutions will be complex or imaginary numbers.

Step 4: Calculate the Complex Solutions

Since the discriminant is negative, apply the quadratic formula to find the complex roots. The formula is \(x = \frac{-b \pm \sqrt{D}}{2a}\). Substitute \(a = 4\), \(b = 1\), and \(D = -47\) into the formula: \(x = \frac{-1 \pm \sqrt{-47}}{8}\). This simplifies to \(x = \frac{-1 \pm i\sqrt{47}}{8}\).

Key Concepts

Discriminant

When working with quadratic equations of the form \(ax^2 + bx + c = 0\), understanding the role of the discriminant is essential. The discriminant \(D\) is calculated using the formula \(D = b^2 - 4ac\). This value helps determine the nature of the roots of the quadratic equation.

• If \(D > 0\), the equation has two distinct real roots.
• If \(D = 0\), there is exactly one real root, often called a repeated or double root.
• If \(D < 0\), which is the case for our equation where \(D = -47\), the roots are complex or imaginary numbers.

In our example, since \(D\) is negative, it tells us immediately that the solutions for the equation \(4x^2 + x + 3 = 0\) will not intersect the x-axis as real numbers, but will instead be complex.

Complex Solutions

Complex solutions arise in quadratic equations when the discriminant is negative. This means there are no real intersections with the x-axis, and the solutions must involve imaginary numbers.
In our example, the equation \(4x^2 + x + 3 = 0\) has a discriminant \(D = -47\), which means the solutions are complex. To understand complex numbers, remember they are composed of a real part and an imaginary part, expressed in the form \(a + bi\), where \(i\) is the imaginary unit with the property that \(i^2 = -1\).
By applying the quadratic formula (more on this below), we substitute the values to get the solutions:
- Real part: \(-\frac{1}{8}\)- Imaginary part: \(\pm \frac{\sqrt{47}}{8}i\)This results in the complex solutions:

• \(x = \frac{-1 + i\sqrt{47}}{8}\)
• \(x = \frac{-1 - i\sqrt{47}}{8}\)

Quadratic Formula

The quadratic formula is a powerful tool for finding solutions to any quadratic equation. It is especially helpful when the equation cannot be easily factored, or when dealing with complex solutions. The formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]For the equation \(4x^2 + x + 3 = 0\), we know from the coefficients that \(a = 4\), \(b = 1\), and \(c = 3\). Substituting these into the quadratic formula, we calculate:\[ x = \frac{-1 \pm \sqrt{-47}}{8} \]The part under the square root, \(\sqrt{-47}\), involves the imaginary unit \(i\), since the square root of a negative number results in an imaginary number. Thus, we rewrite it as \(\pm i\sqrt{47}\), revealing the complex nature of the roots.
The quadratic formula not only delivers the solutions but also encapsulates the discriminant step inherently, showing how each part connects in solving quadratic equations.