Problem 110 - Calculus for Biology and Medicine

Question

First determine whether the solutions of each quadratic equation are real or complex without solving the equation. Then solve the equation. \(-3 x^{2}-x-4=0\)

Step-by-Step Solution

Answer

The solutions are complex: \(x = -\frac{1}{6} \pm \frac{i\sqrt{47}}{6}\).

Step 1: Determine the Discriminant

To find out if the solutions are real or complex, we calculate the discriminant of the quadratic equation. The formula for the discriminant is given by \(\Delta = b^2 - 4ac\), where \(a = -3\), \(b = -1\), and \(c = -4\) in our equation. Substitute these values to get \(\Delta = (-1)^2 - 4(-3)(-4)\).

Step 2: Calculate the Discriminant

Substitute the values into the discriminant formula: \(\Delta = 1 - 48 = -47\). Since the discriminant is less than zero (\(\Delta < 0\)), the solutions are complex numbers.

Step 3: Use the Quadratic Formula

Since the solutions are complex, we will use the quadratic formula: \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\). Substitute \(a = -3\), \(b = -1\), and \(\Delta = -47\) into the formula to get \(x = \frac{-(-1) \pm \sqrt{-47}}{2(-3)}\).

Step 4: Simplify the Quadratic Formula

Simplify the expression: \(x = \frac{1 \pm \sqrt{-47}}{-6}\). Since \(\sqrt{-47} = i\sqrt{47}\), further simplify it to \(x = \frac{1 \pm i\sqrt{47}}{-6}\).

Step 5: Write the Final Solution

The solutions in terms of real and imaginary parts are \(x = -\frac{1}{6} \pm \frac{i\sqrt{47}}{6}\). Therefore, the solutions are complex numbers.

Key Concepts

Discriminant

When you encounter a quadratic equation, the discriminant helps you determine the nature of its solutions. The discriminant is part of the quadratic formula, represented by \( b^2 - 4ac \).

• If the discriminant is positive (\( \Delta > 0 \)), the quadratic equation has two distinct real solutions.
• If the discriminant is zero (\( \Delta = 0 \)), the equation has exactly one real solution, also called a repeated or double root.
• If the discriminant is negative (\( \Delta < 0 \)), the solutions are not real numbers; they are complex numbers.

In our original exercise, the discriminant was \(-47\), a negative number. This indicates the solutions to the quadratic equation are complex, not crossing the x-axis on the graph.

Complex Numbers

Complex numbers are numbers that have both real and imaginary parts. They are typically written in the form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part, with \( i \) being the imaginary unit.

• The imaginary unit \( i \) is defined as the square root of -1, so \( i^2 = -1 \).
• Complex numbers can be added, subtracted, multiplied, and divided like real numbers, accounting for the imaginary part using \( i \).

In the solved equation, we ended up with solutions in terms of complex numbers: \( x = -\frac{1}{6} \pm \frac{i\sqrt{47}}{6} \). This means each solution has a real part \(-\frac{1}{6}\) and an imaginary part \( \pm \frac{\sqrt{47}}{6} i\). Understanding complex numbers is crucial, especially when dealing with negative discriminants.

Quadratic Formula

The quadratic formula is a universal tool for solving any quadratic equation of the form \( ax^2 + bx + c = 0 \). It is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula lets us find the solutions (roots) of the equation by substituting the coefficients \( a \), \( b \), and \( c \).

• The symbols \( \pm \) suggest there will be two solutions, one with a plus and the other with a minus.
• The square root of the discriminant \( \sqrt{b^2 - 4ac} \) determines if the solutions are real or complex.
• For complex solutions, expressed using an imaginary unit \( i \), the process leads to results outside the typical real numbers.

For our exercise, using \( a = -3 \), \( b = -1 \), and a negative discriminant \(-47\), the quadratic formula results in solutions that are expressed in terms of complex numbers. This shows the flexibility of the quadratic formula, even when working with imaginary numbers.