Problem 103 - 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: Write the Equation in Standard Form

To solve the quadratic equation, we first need to write it in the standard quadratic form, which is given by \( ax^2 + bx + c = 0 \). The given equation is \( -x^2 + x + 2 = 0 \). To express it in standard form, we multiply the entire equation by \(-1\) to get \( x^2 - x - 2 = 0 \). Now, it is in the form where \(a = 1\), \(b = -1\), and \(c = -2\).

Step 2: Apply the Quadratic Formula

The quadratic formula for finding the roots of \( ax^2 + bx + c = 0 \) is given by: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substituting the values \( a = 1 \), \( b = -1 \), and \( c = -2 \) from our equation into the formula, we have: \[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4\cdot1\cdot(-2)}}{2\cdot1} \].

Step 3: Calculate the Discriminant

Calculate the discriminant \( b^2 - 4ac \) to determine the nature of the roots. Substituting the values: \( (-1)^2 - 4\cdot1\cdot(-2) = 1 + 8 = 9 \). Since the discriminant is positive, it indicates that the roots are real and distinct.

Step 4: Solve for the Roots

Now, use the value of the discriminant to find the roots. Substitute into the formula: \[ x = \frac{1 \pm \sqrt{9}}{2} \]. Simplifying further: \[ x = \frac{1 \pm 3}{2} \]. This gives two solutions: \( x = \frac{1 + 3}{2} = 2 \) and \( x = \frac{1 - 3}{2} = -1 \). Thus, the solutions are \( x = 2 \) and \( x = -1 \).

Key Concepts

Complex Number System

When solving quadratic equations, sometimes the solutions are not straightforward numbers we are accustomed to, like whole numbers or fractions. Instead, we need to dive into the realm of the complex number system. This is where numbers take the form of \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit that satisfies \( i^2 = -1 \).

• If the discriminant (the expression under the square root in the quadratic formula) is negative, then the solutions are complex numbers.
• If the discriminant is positive, like in our exercise with \( 9 \), the solutions remain in the realm of real numbers.
• Complex numbers allow us to find solutions even when no real number solution exists.

This ability to solve equations fully, regardless of the discriminant's sign, makes the complex number system incredibly powerful.

Quadratic Formula

The quadratic formula is a quintessential tool used to find the solutions of a quadratic equation, which is any equation that can be rearranged to be in the form \( ax^2 + bx + c = 0 \). The roots or solutions of this equation can be calculated using the formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here's how the different parts of the formula work:

• \( -b \) is used to calculate the symmetry around the vertex of the parabola represented by the quadratic equation.
• The term \( \pm \sqrt{b^2 - 4ac} \) determines the nature of the roots, real or complex.
• The denominator \( 2a \) ensures the equation accounts for the leading coefficient of the quadratic term.

By substituting the values of \( a \), \( b \), and \( c \) into the formula, students can systematically find the roots which are the solutions to the quadratic equation.

Standard Form

The standard form of a quadratic equation is like its unique identifier, expressed as \( ax^2 + bx + c = 0 \). It clearly outlines the coefficients \( a \), \( b \), and the constant \( c \), making it easier to apply mathematical techniques accurately.

• \( a \): Coefficient of \( x^2 \), determines the orientation and width of the parabola.
• \( b \): Coefficient of \( x \), which affects the vertex's horizontal position relative to the y-axis.
• \( c \): The constant term, gives the parabola's y-intercept.

Rewriting equations into this standard form as a first step is crucial in solving quadratics, as it directly prepares the equation for the application of the quadratic formula or other strategies, such as factoring or completing the square.

Discriminant

The discriminant is a critical part of understanding the solutions of a quadratic equation. It is the part under the square root in the quadratic formula: \( b^2 - 4ac \). The discriminant tells us key information:

• If \( b^2 - 4ac > 0 \), the equation has two distinct real roots, as seen in our exercise.
• If \( b^2 - 4ac = 0 \), there is exactly one real root, meaning the parabola touches the x-axis at just one point.
• If \( b^2 - 4ac < 0 \), the solutions are complex and come in conjugate pairs, indicating that the parabola does not cross the x-axis.

Knowing the discriminant helps predict the nature of the roots and guides us towards the most efficient method to solve the quadratic equation.