Problem 22 - Basic Technical Mathematics with Calculus

Question

solve the given quadratic equations, using the quadratic formula. Exercises \(5-8\) are the same as Exercises \(11-14\) of Section 7.2. $$2 d(d-2)=-7$$

Step-by-Step Solution

Answer

The solutions are \(d = 1 \pm \frac{\sqrt{10}i}{2}\). The roots are complex numbers.

Step 1: Rewrite the equation

First, we want to express the quadratic equation in the standard form, which is \(ax^2 + bx + c = 0\). Start by expanding and rearranging the given equation \(2d(d-2) = -7\). Expand the left side to get \(2d^2 - 4d\). Then, add 7 to both sides to set the equation to zero: \(2d^2 - 4d + 7 = 0\).

Step 2: Identify coefficients

Identify the coefficients \(a\), \(b\), and \(c\) in the quadratic equation \(2d^2 - 4d + 7 = 0\). Here, \(a = 2\), \(b = -4\), and \(c = 7\).

Step 3: Apply the quadratic formula

The quadratic formula is given by \(d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute the values of \(a\), \(b\), and \(c\) into the formula: \(d = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(2)(7)}}{2(2)}\).

Step 4: Calculate the discriminant

Calculate the discriminant inside the square root: \((-4)^2 - 4 \times 2 \times 7 = 16 - 56 = -40\). Since the discriminant is negative, the roots are complex numbers.

Step 5: Simplify under the square root

Simplify \(\sqrt{-40}\). Rewrite it as \(\sqrt{40} \cdot i\), where \(i\) is the imaginary unit. Since \(40 = 4 \times 10\), find \(\sqrt{40} = 2\sqrt{10}\). Thus, \(\sqrt{-40} = 2\sqrt{10}i\).

Step 6: Final solution with complex roots

Substitute back the simplified square root into the quadratic formula: \(d = \frac{4 \pm 2\sqrt{10}i}{4}\). Simplify the fraction: \(d = 1 \pm \frac{\sqrt{10}i}{2}\). There are two complex solutions: \(d = 1 + \frac{\sqrt{10}i}{2}\) and \(d = 1 - \frac{\sqrt{10}i}{2}\).

Key Concepts

Complex Numbers

Complex numbers are numbers that have two parts: a real part and an imaginary part. They are typically written in the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part. The imaginary unit \(i\) is defined as \(\sqrt{-1}\), which allows us to handle square roots of negative numbers. This inclusion of an imaginary part makes complex numbers a complete number system.

• Real part: In our scenario, \(1\) is the real component.
• Imaginary part: Here, \(\frac{\sqrt{10}i}{2}\) is the imaginary component.
• Complex conjugate: For a complex number \(a + bi\), its complex conjugate is \(a - bi\). In our solutions, the complex conjugates are \(1 + \frac{\sqrt{10}i}{2}\) and \(1 - \frac{\sqrt{10}i}{2}\).

Complex numbers are crucial in solving quadratic equations with negative discriminants, leading us to solutions that "extend" the real number system to ensure that all quadratic equations have solutions.

Discriminant

The discriminant, denoted as \(b^2 - 4ac\), is a critical part of the quadratic formula \(d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). It helps us determine the nature of the roots of a quadratic equation.

• Positive Discriminant: When \(b^2 - 4ac > 0\), the quadratic has two distinct real roots.
• Zero Discriminant: When \(b^2 - 4ac = 0\), it has exactly one real root (a double root).
• Negative Discriminant: In our example, \(b^2 - 4ac\) was \(-40\), indicating two complex roots. This happens because the square root of a negative number introduces an imaginary component, leading to complex solutions.

The discriminant is a concise way to quickly assess what kind of solutions a quadratic equation will yield, without needing to solve the entire equation.

Imaginary Numbers

Imaginary numbers arise when we take the square root of a negative number. Since the square of any real number is always positive, mathematicians defined an imaginary unit \(i\) such that \(i^2 = -1\) to handle these scenarios.

• Imaginary Unit: \(i = \sqrt{-1}\).
• Using \(i\): When the discriminant is negative, as in our problem \(-40\), we rewrite it using \(i\). Thus, \(\sqrt{-40} = \sqrt{40}i = 2\sqrt{10}i\).
• Operations: Imaginary numbers follow similar operation rules as real numbers, with additional rules for multiplication (\(i \times i = i^2 = -1\)).

Imaginary numbers, when combined with real numbers, form complex numbers, enabling us to solve equations that do not have real number solutions, thereby expanding our mathematical toolkit.