Problem 100 - Calculus for Biology and Medicine

Question

Solve each quadratic equation in the complex number system. \(x^{2}+x+1=0\)

Step-by-Step Solution

Answer

The roots are complex: \(x = \frac{-1 + i\sqrt{3}}{2}\) and \(x = \frac{-1 - i\sqrt{3}}{2}\).

Step 1: Identify the Quadratic Equation Structure

The given equation is already in the standard form of a quadratic equation, which is \(ax^2 + bx + c = 0\). For this equation, \(a = 1\), \(b = 1\), and \(c = 1\).

Step 2: Calculate the Discriminant

Use the discriminant formula \(D = b^2 - 4ac\). Substitute \(a = 1\), \(b = 1\), and \(c = 1\) into the formula to find \(D = 1^2 - 4 \times 1 \times 1 = 1 - 4 = -3\). The negative discriminant indicates that the roots are complex.

Step 3: Apply the Quadratic Formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{D}}{2a}\). Substitute \(b = 1\), \(D = -3\), and \(a = 1\) into the formula to find the roots.

Step 4: Solve for Complex Roots

Substitute the values: \(x = \frac{-1 \pm \sqrt{-3}}{2}\). Simplify the square root of the negative number: \(\sqrt{-3} = i\sqrt{3}\). Thus, the equation becomes \(x = \frac{-1 \pm i\sqrt{3}}{2}\).

Step 5: Write the Final Solution

The solutions to the quadratic equation are \(x = \frac{-1 + i\sqrt{3}}{2}\) and \(x = \frac{-1 - i\sqrt{3}}{2}\). These are the roots expressed in the complex number system.

Key Concepts

Complex Numbers

Mathematics isn't confined to the world of real numbers. It extends its reach to complex numbers, which are essential in solving many algebraic equations. Complex numbers are of the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is an imaginary unit. The imaginary unit \(i\) is unique because \(i^2 = -1\). This means that when dealing with square roots of negative numbers, complex numbers become crucial. For example, \(\sqrt{-3}\) can be expressed as \(i\sqrt{3}\). Complex numbers are often visualized on a two-dimensional plane known as the complex plane. In this representation, the real part, \(a\), lies on the horizontal axis, while the imaginary part, \(bi\), is plotted on the vertical axis. Understanding complex numbers helps solve many types of equations that have no real solutions, like the quadratic equation \(x^2 + x + 1 = 0\), due to its negative discriminant. In this case, the roots are complex, demonstrating the essential role complex numbers play in mathematics.

Discriminant

The discriminant is a key tool in quadratic equations, easily calculated using the formula \(D = b^2 - 4ac\). This single value provides important information about the nature of the roots of a quadratic equation \(ax^2 + bx + c = 0\).

• When \(D > 0\), the quadratic equation has two distinct real roots.
• If \(D = 0\), the equation has exactly one real root, or a double root.
• When \(D < 0\), as seen in this exercise with \(D = -3\), it indicates that the equation has two complex roots. This is because the square root of a negative number will involve the imaginary unit \(i\).

For the equation \(x^2 + x + 1 = 0\), since \(D = -3\), the roots are complex. The discriminant not only signals the type of roots but also paves the way for using the quadratic formula to find these roots.

Quadratic Formula

The quadratic formula is a powerful and versatile tool useful for finding the roots of any quadratic equation \(ax^2 + bx + c = 0\). It is given by:\[ x = \frac{-b \pm \sqrt{D}}{2a} \]Here, \(b\) is the coefficient of \(x\), \(a\) is the coefficient of \(x^2\), \(c\) is the constant term, and \(D\) is the discriminant. The "±" symbol implies the presence of two solutions, hence two roots.For the equation \(x^2 + x + 1 = 0\), substituting the values, \(a = 1\), \(b = 1\), and \(D = -3\), into the quadratic formula, we find:\[ x = \frac{-1 \pm i\sqrt{3}}{2} \]Since the discriminant is negative, the square root part becomes imaginary, resulting in complex solutions. This illustrates how the quadratic formula not only handles real numbers but also adeptly manages solutions within the complex number system, making it a fundamental tool in algebra.