Problem 101 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The solutions are \(x = 2\) and \(x = -1\).

Step 1: Rearrange the Equation

First, we need to rewrite the given equation into the standard form of a quadratic equation, which is \( ax^2 + bx + c = 0 \). Given the equation is \(-x^2 + x + 2 = 0\), multiply the whole equation by \(-1\) to get: \(x^2 - x - 2 = 0\).

Step 2: Identify Coefficients

Now identify the coefficients from the standard form equation \(x^2 - x - 2 = 0\). Here, \(a = 1\), \(b = -1\), and \(c = -2\). These are the coefficients to use in the quadratic formula.

Step 3: Apply the Quadratic Formula

Use the quadratic formula to find the roots of the equation. The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\\]Substitute \(a = 1\), \(b = -1\), and \(c = -2\) into the formula:\[x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \times 1 \times (-2)}}{2 \times 1}\\]

Step 4: Simplify the Expression

Simplify the expression inside the square root (known as the discriminant):\[x = \frac{1 \pm \sqrt{1 + 8}}{2}\\]\(8\) comes from calculating \(4 \times 1 \times (-2)\). Now simplify further:\[x = \frac{1 \pm \sqrt{9}}{2}\\]

Step 5: Calculate the Roots

Calculate the roots by solving:\[x = \frac{1 \pm 3}{2}\\]This gives two solutions:1. \(x = \frac{1 + 3}{2} = 2\)2. \(x = \frac{1 - 3}{2} = -1\)

Key Concepts

Complex Number System

In mathematics, the concept of the complex number system expands our understanding of numbers beyond the real number line. Complex numbers have a real part and an imaginary part. This system is especially useful when dealing with solutions to equations that don't have real number solutions. For example, in quadratic equations where the polynomial doesn't cross the x-axis, solutions would involve complex numbers.
Complex numbers are represented as

• The format: \( a + bi \)
• Where \( a \) (the real part) is a real number
• \( b \) (the imaginary part) is a real number multiplied by \( i \), the imaginary unit

Here, \( i \) is defined as \( i^2 = -1 \). Encountering complex solutions in a quadratic equation is not unusual as it can occur when the discriminant is negative. The solution process is similar, except you will need to factor in the imaginary unit \( i \) during calculation.

Quadratic Formula

The quadratic formula is a remarkable tool used to find the roots of any quadratic equation, expressed in the standard form:
\( ax^2 + bx + c = 0 \). This formula provides the exact roots of the equation and is given by:

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

To employ the quadratic formula, identify the coefficients \( a \), \( b \), and \( c \) from your equation. Substitute these values into the formula to solve for \(x\). You will get two solutions because of the \( \pm \) symbol, representing the two possible roots of the quadratic equation.Each step involves:

• Substitute the coefficients into the formula.
• Calculate the discriminant, \( b^2 - 4ac \), to understand the nature of the roots.
• Simplify the expression inside the square root, then proceed with calculation.

Using the quadratic formula helps in finding solutions to quadratic equations, be they real or complex.

Discriminant

The discriminant is a pivotal part of the quadratic formula, providing insights into the type of solutions you can expect. It's the expression under the square root in the quadratic formula:

• \( b^2 - 4ac \)

The value of the discriminant informs about the nature of the roots:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If the discriminant is zero, there is exactly one real root, represented by a repeated root.
• If the discriminant is negative, the equation will have two complex roots.

Understanding the discriminant is crucial because it prepares you for what type of arithmetic calculations you will engage with. For the quadratic equation \( x^2 - x - 2 = 0 \), the discriminant is calculated as \( 1 - 4(-2) = 9 \), indicating two distinct real roots.

Algebraic Solutions

Solving quadratic equations through algebraic solutions involves different methods depending on what is most feasible and efficient. These methods include:

• Factoring, which splits the quadratic into two binomial expressions if easily factorable.
• Completing the square, useful when the quadratic is perfect for converting into a square from its original form.
• Using the quadratic formula for general solutions, effective when other methods are cumbersome.

In the context of complex numbers, the quadratic formula is generally the best option. It allows for solving when simpler algebraic methods aren't viable. It's accurate in getting exact roots, making it a staple technique when dealing with quadratics in a complex number system. For our equation \( x^2 - x - 2 = 0 \), applying these methods confirms it can be straightforwardly solved using the quadratic formula, yielding the solutions \( x = 2 \) and \( x = -1 \).