Problem 107 - 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 real: \( x = \frac{4}{3} \) and \( x = -1 \).

Step 1: Identify the Coefficients

A quadratic equation is generally written in the form \( ax^2 + bx + c = 0 \). Identify the coefficients \( a \), \( b \), and \( c \) for the equation \( 3x^2 - x - 4 = 0 \). Here, \( a = 3 \), \( b = -1 \), and \( c = -4 \).

Step 2: Calculate the Discriminant

The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula \( D = b^2 - 4ac \). Substitute the identified values into this formula: \( D = (-1)^2 - 4 \times 3 \times (-4) \).

Step 3: Evaluate the Discriminant

Compute the value of the discriminant: \( D = 1 + 48 = 49 \). Since \( D > 0 \), the quadratic equation has two distinct real solutions.

Step 4: Use the Quadratic Formula

To find the solutions, use the quadratic formula \( x = \frac{-b \pm \sqrt{D}}{2a} \). Substitute \( a = 3 \), \( b = -1 \), and \( D = 49 \) into the formula: \( x = \frac{1 \pm \sqrt{49}}{6} \).

Step 5: Solve for the Solutions

Compute the solutions using the quadratic formula: \( x = \frac{1 + 7}{6} = \frac{8}{6} = \frac{4}{3} \) and \( x = \frac{1 - 7}{6} = \frac{-6}{6} = -1 \).

Key Concepts

Understanding the Discriminant

The concept of the discriminant is crucial when working with quadratic equations. The discriminant helps us determine the nature of the solutions, without actually solving the equation itself. It's a very handy tool!
For a quadratic equation in the form of \( ax^2 + bx + c = 0 \), the discriminant is calculated as follows:

• \( D = b^2 - 4ac \)

The value of \( D \) tells us the nature of the roots of the quadratic equation:

• If \( D > 0 \), the equation has two distinct real solutions.
• If \( D = 0 \), the equation has exactly one real solution, known as a repeated or double root.
• If \( D < 0 \), the equation has two complex solutions (which are not real).

In the given exercise, the discriminant turned out to be 49, which is greater than 0. Thus, we could predict there would be two distinct real solutions.

What Are Real Solutions?

Real solutions of a quadratic equation refer to the values of \( x \) that solve the equation and are real numbers. These are the points where the graph of the quadratic equation, a parabola, crosses the x-axis.
Understanding the nature of solutions is essential for plotting the graph of a quadratic function, as well as in practical applications. When the discriminant is positive, as it was in our exercise (\( D = 49 \)), it signals that the parabola will intersect the x-axis at two points. This means that there are two distinct x-values, or real solutions, where the equation \( ax^2 + bx + c = 0 \) holds true.
In a real-world context, real solutions might represent measurable outcomes, such as the time a ball takes to hit the ground when tossed into the air, or the dimensions needed to solve an engineering problem.

The Quadratic Formula Made Simple

The quadratic formula is a powerful method we use to find the solutions to any quadratic equation. Once you've determined the nature of the solutions using the discriminant, the quadratic formula can help you find the exact solutions quickly.
The formula is structured as:

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

Let's break it down with an example. In the given equation \( 3x^2 - x - 4 = 0 \), we identified \( a = 3 \), \( b = -1 \), and the discriminant \( D = 49 \).
Substituting these into the quadratic formula results in:

• \( x = \frac{1 \pm \, \sqrt{49}}{6} \)

The formula allows us to solve for both potential solutions at once, with \( + \) and \( - \) giving us the distinct real solutions:

• \( x = \frac{4}{3} \)
• \( x = -1 \)

Whenever you encounter a quadratic equation, this formula is your go-to tool to find the solution, ensuring you have the skills needed to tackle these problems with confidence!