Problem 39 - Algebra and Trigonometry with Analytic Geometry

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 Quadratic Equation

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

Step 2: Calculate the Discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(b^2 - 4ac\). Substitute the values, \(b = -6\), \(a = 1\), and \(c = 13\) into the formula to find the discriminant: \((-6)^2 - 4 \times 1 \times 13 = 36 - 52 = -16\).

Step 3: Analyze the Discriminant

The discriminant \(\Delta = -16\) is less than zero, indicating that the quadratic equation has two complex roots.

Step 4: Find the Roots Using the Quadratic Formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Because the discriminant is negative, the roots will be complex numbers. Substitute \(a = 1\), \(b = -6\), and \(c = 13\) into the formula: \[x = \frac{-(-6) \pm \sqrt{-16}}{2 \times 1} = \frac{6 \pm 4i}{2} = 3 \pm 2i\].

Step 5: State the Solutions

The solutions to the equation \(x^2 - 6x + 13 = 0\) are the complex numbers \(x = 3 + 2i\) and \(x = 3 - 2i\).

Key Concepts

Discriminant

The discriminant is an essential part of solving quadratic equations. It is the expression found under the square root symbol in the quadratic formula, specifically \(b^2 - 4ac\). By calculating the discriminant, we can determine the nature of the roots of the equation.
For any quadratic equation of the form \(ax^2 + bx + c = 0\), follow these insights:

• If the discriminant is positive, \(b^2 - 4ac > 0\), there are two distinct real roots.
• If it is zero, \(b^2 - 4ac = 0\), there is exactly one real root, or the roots are repeated.
• If the discriminant is negative, \(b^2 - 4ac < 0\), the roots involve complex numbers, as is the case in our equation.

Understanding the discriminant helps us predict whether the solutions are real or complex without even fully solving the quadratic equation.

Example from Our Equation

In the given equation \(x^2 - 6x + 13 = 0\), the discriminant \(b^2 - 4ac\) was computed as \(-16\). This negative value tells us immediately that the solutions are complex roots.

Complex Roots

Complex roots arise when the discriminant of a quadratic equation is less than zero. In simpler terms, when you end up with a negative number under the square root in the quadratic formula, the solutions can't be real numbers. Instead, they are complex numbers that include the imaginary unit \(i\), where \(i = \sqrt{-1}\).
These roots reflect a beautiful aspect of mathematics where equations extend beyond real numbers.

• Complex roots always come in conjugate pairs, like \(a + bi\) and \(a - bi\), ensuring symmetry in the complex plane.
• This symmetry plays an essential role in many areas of math, including geometry and signal processing.

Finding Complex Roots

Using the quadratic formula, if the discriminant is negative, your calculated roots will appear as, for instance, \(3 \pm 2i\), as seen in our example. Here, each part (real and imaginary) is calculated directly from the formula, ensuring precise answers.

Quadratic Formula

The quadratic formula is a historic and reliable method for solving any quadratic equation. It's directly derived from the process of completing the square, a technique for transforming standard-form quadratics into vertex form.
The formula is written as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This versatile formula allows you to solve for \(x\) by only inputting the coefficients \(a\), \(b\), and \(c\) from your quadratic equation \(ax^2 + bx + c = 0\).

• First, compute the discriminant \(b^2 - 4ac\).
• Plug the values into the formula.
• Solve for \(x\) using standard algebraic operations.

Application in Our Equation

For \(-6x + 13 = 0\), substituting \(a = 1, b = -6, c = 13\) into the quadratic formula offers the solutions \(x = 3 + 2i\) and \(x = 3 - 2i\). The formula's beauty lies in its simplicity, whether dealing with real or complex roots.