Problem 40 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. \(39-56:\) Find the solutions of the equation. $$ x^{2}-2 x+26=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 1 + 5i\) and \(x = 1 - 5i\).

Step 1: Identify the Equation Type

The given equation is a quadratic equation, which is of the form \(ax^2 + bx + c = 0\). Here, \(a = 1\), \(b = -2\), and \(c = 26\).

Step 2: Calculate the Discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(D = b^2 - 4ac\). Substitute \(b = -2\), \(a = 1\), and \(c = 26\) into the formula, so: \[D = (-2)^2 - 4 \times 1 \times 26 = 4 - 104 = -100.\]

Step 3: Interpret the Discriminant

The discriminant \(D = -100\) is negative, indicating that the quadratic equation has two complex solutions (no real solutions). These solutions will be complex conjugates.

Step 4: Use the Quadratic Formula

For a quadratic equation \(ax^2 + bx + c = 0\), the solutions are given by the quadratic formula: \[x = \frac{-b \pm \sqrt{D}}{2a}.\]Substitute \(b = -2\), \(a = 1\), and \(D = -100\) into the formula: \[ x = \frac{-(-2) \pm \sqrt{-100}}{2 \times 1} = \frac{2 \pm \sqrt{100i^2}}{2}. \]

Step 5: Simplify the Solutions

Calculate \(\sqrt{-100} = 10i\). Substitute back and simplify: \[x = \frac{2 \pm 10i}{2}.\] This gives the solutions: \[x = 1 \pm 5i.\]

Step 6: State the Solutions

The equation \(x^2 - 2x + 26 = 0\) has the solutions \(x = 1 + 5i\) and \(x = 1 - 5i\).

Key Concepts

Complex Solutions

Quadratic equations can sometimes have solutions that are not real numbers. These are called complex solutions, and they arise whenever the equation’s discriminant is negative. Complex solutions consist of a real part and an imaginary part, symbolically represented using the imaginary unit "i", where \( i^2 = -1 \). In the case of our example, when solving the equation \(x^2 - 2x + 26 = 0\), we determined that it yields complex solutions because of the negative discriminant. This means the solutions will be of the form \(a \pm bi\).

Here, \(1 + 5i\) and \(1 - 5i\) are complex conjugates. They have the same real part but opposite imaginary parts. Conjugates are important because they nicely cancel out imaginary components when used in computation, maintaining real results in contexts where needed.

Quadratic Formula

The quadratic formula is a useful tool for finding the solutions of a quadratic equation, particularly when it is not easily factored. For any equation of the form \(ax^2 + bx + c = 0\), the solutions can be found using the formula:

\[x = \frac{-b \pm \sqrt{D}}{2a}\]

where \(D\) is the discriminant \(b^2 - 4ac\). This formula allows us to plug in any values for \(a\), \(b\), and \(c\) and solve for \(x\), regardless of whether the solutions are real or complex.

In our example, substituting \(b = -2\), \(a = 1\), and \(D = -100\) into the quadratic formula gave us the complex solutions \(1 + 5i\) and \(1 - 5i\). The nested plus-minus symbol indicates that the solutions typically come in pairs, often complex conjugates when the discriminant is negative.

Discriminant

The discriminant in a quadratic equation is a key determining factor in understanding the nature of the solutions. Given by the formula \(D = b^2 - 4ac\), it tells us whether the solutions are real or complex. Here’s what the discriminant indicates:

• If \(D > 0\), there are two distinct real solutions.
• If \(D = 0\), there is exactly one real solution, also known as a repeated root.
• If \(D < 0\), the solutions are complex conjugates, indicating no real solutions.

In the equation \(x^2 - 2x + 26 = 0\), we calculated \(D = -100\), which is less than zero, thus leading to complex solutions. By understanding and applying the discriminant, we gain insight into the characteristics of the equation's solutions._END