Problem 42 - Algebra and Trigonometry with Analytic Geometry

Question

Find the solutions of the equation $$ x^{2}+8 x+17=0 $$

Step-by-Step Solution

Answer

The solutions are \(-4 + i\) and \(-4 - i\).

Step 1: Identify the Type of Equation

The equation given is a quadratic equation because the highest power of the variable \(x\) is 2. It is of the form \(ax^2 + bx + c = 0\) where \(a = 1\), \(b = 8\), and \(c = 17\).

Step 2: Calculate the Discriminant

To find the nature of the roots, calculate the discriminant \(D\), which is given by \(b^2 - 4ac\). Substitute \(a = 1\), \(b = 8\), \(c = 17\) into the formula: \(D = (8)^2 - 4 \cdot 1 \cdot 17 = 64 - 68 = -4\). The discriminant is negative.

Step 3: Interpret the Discriminant

Since the discriminant \(D\) is negative, the quadratic equation has two complex roots. Complex roots occur in conjugate pairs.

Step 4: Use the Quadratic Formula

The roots of the quadratic equation can be found using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute \(a = 1\), \(b = 8\), and \(D = -4\) into the formula to get: \[x = \frac{-8 \pm \sqrt{-4}}{2 \cdot 1}\].

Step 5: Simplify the Roots

Since \(\sqrt{-4} = 2i\), replace it in the formula: \[x = \frac{-8 \pm 2i}{2}\]. Simplify by dividing each part by 2: \[x = -4 \pm i\]. Thus, the roots are \(-4 + i\) and \(-4 - i\).

Key Concepts

Understanding the Discriminant

In a quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant helps determine the nature of the roots. The discriminant is calculated using the formula \(D = b^2 - 4ac\). This small yet powerful number tells us the type of roots the quadratic equation will have.

There are three possibilities based on the value of the discriminant:

• If \(D > 0\), the equation has two distinct real roots.
• If \(D = 0\), the equation has exactly one real root; this is often called a double root.
• If \(D < 0\), the equation has two complex roots.

In the given equation \(x^2 + 8x + 17 = 0\), the discriminant is calculated as \(D = 64 - 68 = -4\), which is negative. This negative value indicates that the equation does not have real roots. Instead, the roots are complex numbers.

Complex Roots in Quadratic Equations

Complex roots occur when the discriminant \(D\) is less than zero. Complex roots always come in conjugate pairs, reflecting a harmony in their real and imaginary components. A complex number can be expressed as \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part.

In our example, the discriminant is \(-4\), which results in complex roots. When simplified using the quadratic formula, the roots turn out to be \(-4 + i\) and \(-4 - i\).

These are conjugate pairs because:

• They have the same real part \(-4\).
• The imaginary parts \(+i\) and \(-i\) are negatives of each other.

It's essential to grasp this pairing concept because it reflects the symmetry in complex roots, indicating they are reflections about the real number line.

The Quadratic Formula Explained

The quadratic formula is a universal solution to find the roots of any quadratic equation \(ax^2 + bx + c = 0\). The formula is:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

This formula is derived from completing the square on the general quadratic equation. Let's break down how it was applied in our problem:

• First, identify \(a = 1\), \(b = 8\), and \(c = 17\).
• Calculate the discriminant \(D = b^2 - 4ac\) which equals \(-4\).
• Substitute these values into the formula:

\[x = \frac{-8 \pm \sqrt{-4}}{2 \times 1}\]

This simplifies to \(x = -4 \pm i\), after considering that \(\sqrt{-4} = 2i\). Thus, the roots are \(-4 + i\) and \(-4 - i\), clearly showing the utilization of the quadratic formula in handling complex roots.

The formula not only provides a method for calculating roots but also highlights the importance of analyzing the discriminant for predicting root types.