Problem 39 - Precalculus: Functions and Graphs

Question

Find the solutions of the equation. $$x^{2}-6 x+13=0$$

Step-by-Step Solution

Answer

The solutions are \( x = 3 + 2i \) and \( x = 3 - 2i \).

Step 1: Identify the equation type

The given equation is a quadratic equation of the form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = -6 \), and \( c = 13 \).

Step 2: Check the discriminant

The discriminant \( \Delta \) is computed using the formula \( \Delta = b^2 - 4ac \). Calculate \( (-6)^2 - 4 \cdot 1 \cdot 13 = 36 - 52 = -16 \). Since the discriminant is negative, the quadratic equation has two complex solutions.

Step 3: Solve using the quadratic formula

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substituting the identified values gives: \( x = \frac{-(-6) \pm \sqrt{-16}}{2 \cdot 1} \).

Step 4: Simplify the expressions

First, simplify the expression inside the square root: \( \sqrt{-16} = 4i \), where \( i \) is the imaginary unit. Now, substitute back into the quadratic formula to get \( x = \frac{6 \pm 4i}{2} \).

Step 5: Final calculation for solutions

Divide each term by 2: \( x = 3 \pm 2i \). Therefore, the solutions of the quadratic equation are \( x = 3 + 2i \) and \( x = 3 - 2i \).

Key Concepts

Complex Solutions

In the world of quadratic equations, having complex solutions might initially seem daunting. However, they are simply solutions that involve imaginary numbers. Imaginary numbers arise when we take the square root of a negative number, which is not possible in the set of real numbers. To handle this, we use the imaginary unit 'i', which is defined as \( i = \sqrt{-1} \).
When a quadratic equation has complex solutions, it means the equation doesn't intersect the x-axis on a graph of real numbers. Instead, complex solutions indicate a shift into the complex plane.

• Complex solutions are often represented as \( a + bi \), where \( a \) and \( b \) are real numbers.
• In the exercise above, the solutions \( x = 3 + 2i \) and \( x = 3 - 2i \) showcase the presence of imaginary parts, emphasized by 'i'.
• These solutions reflect a balance in the equation, guaranteeing that when mathematically calculated, the left-hand side equals zero, meeting the fundamental property of quadratic equations.

Discriminant

The discriminant is a critical element in solving quadratic equations, as it helps us understand the nature of the solutions without actually solving the equation fully.
The discriminant \( \Delta \) is calculated using the formula:\[\Delta = b^2 - 4ac\]where \( a \), \( b \), and \( c \) are the coefficients of the quadratic equation \( ax^2 + bx + c = 0 \). This value can tell us much about the solutions:

• A positive discriminant indicates two distinct real solutions.
• A zero discriminant suggests exactly one real solution.
• A negative discriminant reveals two complex solutions, infused with imaginary numbers, as in the provided example with a discriminant of \(-16\).

Observing the discriminant helps anticipate the nature of the solutions and adjust our methods accordingly.

Quadratic Formula

The quadratic formula is a dependable tool in the realm of algebra, allowing us to find solutions to any quadratic equation. The formula is written as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This is particularly useful when other factoring methods seem impractical, such as when dealing with irrational or complex numbers. Here's why it's crucial:

• The formula is versatile and straightforward, suitable for any quadratic equation, regardless of its simplicity or complexity.
• It combines the coefficients and the discriminant to provide solutions directly, where the "+/-" indicates the potential for two solutions.

In our example, this formula helped transition the step of having a negative discriminant to finding meaningful solutions, even when that meant turning to complex numbers. Using the quadratic formula, we calculated that \( x = \frac{6 \pm 4i}{2} \), simplifying further to reveal the complex solutions \( x = 3 + 2i \) and \( x = 3 - 2i \).
Mastering the quadratic formula enables precise and confident problem-solving across diverse quadratic equations.