Problem 104 - Calculus for Biology and Medicine

Question

Solve each quadratic equation in the complex number system. \(-2 x^{2}+x+3=0\)

Step-by-Step Solution

Answer

The solutions are \(x = -1\) and \(x = \frac{3}{2}\).

Step 1: Identify coefficients

First, identify the coefficients from the quadratic equation \(-2x^2 + x + 3 = 0\). Here, the coefficient \(a = -2\), \(b = 1\), and \(c = 3\).

Step 2: Calculate the discriminant

The discriminant \(D\) is calculated using the formula \(D = b^2 - 4ac\). Substitute in the values to get \(D = 1^2 - 4(-2)(3) = 1 + 24 = 25\).

Step 3: Use the quadratic formula

Since the discriminant is positive, use the quadratic formula.The quadratic formula is \(x = \frac{-b \pm \sqrt{D}}{2a}\).Substitute in the values: \(x = \frac{-(1) \pm \sqrt{25}}{2(-2)}\).

Step 4: Simplify using the quadratic formula

First calculate \(\sqrt{25} = 5\). Then substitute back into the formula to get:\[x = \frac{-1 \pm 5}{-4}\]. This breaks down further into two possible solutions:1. \[x_1 = \frac{-1 + 5}{-4} = \frac{4}{-4} = -1\]2. \[x_2 = \frac{-1 - 5}{-4} = \frac{-6}{-4} = \frac{3}{2}\].

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation that involves a variable raised to the power of two. It typically takes the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The highest exponent in a quadratic equation is 2, which is why it's called "quadratic," derived from the Latin word "quadratus," meaning square.

To solve a quadratic equation, you can utilize various methods such as factoring, completing the square, or the quadratic formula, which we will explore shortly. These equations can have real or complex solutions, depending on the value of the discriminant, a key factor in determining the nature of the roots.

Understanding how to identify the coefficients correctly is the first step in solving quadratic equations. In any quadratic equation of the form \( ax^2 + bx + c = 0 \), \( a \) is the coefficient of \( x^2 \), \( b \) is the coefficient of \( x \), and \( c \) is the constant term. For example, in the equation \(-2x^2 + x + 3 = 0\), the values are \( a = -2 \), \( b = 1 \), and \( c = 3 \).

Discriminant

The discriminant is a part of the quadratic formula and plays a crucial role in determining the nature of the roots of a quadratic equation. Denoted by \( D \), the discriminant is calculated using the formula \( D = b^2 - 4ac \).

- **If \( D > 0 \)**: The quadratic equation has two distinct real roots. This indicates the parabola intersects the x-axis at two points.
- **If \( D = 0 \)**: The quadratic equation has exactly one real root (a repeated root). The parabola just touches the x-axis.
- **If \( D < 0 \)**: The quadratic equation has two complex roots, which means the parabola does not intersect the x-axis at all.

In our original exercise, the discriminant was calculated as \( D = 25 \). Since \( D \) is greater than 0, it confirms the presence of two distinct real solutions for the equation \(-2x^2 + x + 3 = 0\).

Quadratic Formula

The quadratic formula is a widely-used tool that provides the solutions to any quadratic equation. The formula is given by:

• \( x = \frac{-b \pm \sqrt{D}}{2a} \)

where \( D \) is the discriminant \( b^2 - 4ac \).

This formula is particularly useful because it allows you to bypass more complex methods like factoring when the equation does not easily factorize.

In the context of our exercise, we started with \(-2x^2 + x + 3 = 0\), calculated the discriminant \( D = 25 \) and used the quadratic formula:

• Substituting into the formula, we applied \( x = \frac{-1 \pm \sqrt{25}}{2(-2)} \).
• After calculation, we found two solutions: \( x_1 = -1 \) and \( x_2 = \frac{3}{2} \), which are real numbers due to the positive discriminant.

The plus-minus symbol "\( \pm \)" in the formula signifies both the addition and subtraction scenarios, leading to two separate results or "roots." Being adept in using the quadratic formula efficiently means you can solve any quadratic equation, even if its roots are not rational numbers.