Problem 93 - Algebra for College Students

Question

Find all real or imaginary solutions to each equation. Use the method of your choice. $$x^{2}-6 x+10=0$$

Step-by-Step Solution

Answer

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

Step 1 - Identify the quadratic equation

The given equation is in the form of a quadratic equation: \[\begin{equation} \begin{cases} a = 1 \ b = -6 \ c = 10 \ x^{2}-6 x+10=0 \ \text{We will proceed to solve it using the quadratic formula.} \ \text{Recall that the quadratic formula is:} \ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}onumber \ \text{In this equation,} \ b^2 - 4ac \text{is the discriminant.} \ \text{Now we substitute a, b, and c with their values.} \end{cases} \end{equation}\]

Step 2 - Calculate the discriminant

Calculate the discriminant using \(b^2-4ac\): \[\begin{equation} \begin{cases} b^2-4ac = (-6)^2 - 4(1)(10) = 36 - 40 = -4 \ \text{Since the discriminant is negative,} \ \text{we will have complex solutions.} \ \text{Now, proceed to find the solutions using the quadratic formula.} \end{cases} \end{equation}\]

Step 3 - Apply the quadratic formula

Substitute the values into the quadratic formula: \[\begin{equation} \begin{cases} x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} = \frac{6 \pm \sqrt{-4}}{2(1)} = \frac{6 \pm \sqrt{4i^2}}{2} = \frac{6 \pm \2i}{2}onumber \ x = 3 \pm \ i \ \text{Therefore, the solutions are} \ x = 3 + i \ x = 3 - i \ \text{which are complex numbers.} \end{cases} \end{equation}\]

Key Concepts

Complex Numbers

Complex numbers are numbers that include a real part and an imaginary part. The imaginary part is represented using the imaginary unit, denoted as \( i \), where \( i^2 = -1 \). For example, \(3 + i\) and \(3 - i\) are complex numbers. These arise naturally when solving quadratic equations with a negative discriminant, leading to solutions that include the square root of a negative number.

Quadratic Formula

The quadratic formula is a method for solving quadratic equations of the form \(ax^2 + bx + c = 0\). The solutions for \(x\) can be found using the formula: \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \). This formula is derived from completing the square and is applicable to any quadratic equation. It's crucial to correctly identify the coefficients \(a\), \(b\), and \(c\) from the given equation. In our example, we used \(a = 1\), \(b = -6\), and \(c = 10\).

Discriminant

The discriminant is the part of the quadratic formula under the square root, noted as \( b^2 - 4ac \). It determines the nature of the roots of the quadratic equation. There are three cases:

• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, the equation has exactly one real root (a repeated root).
• If it is negative, the equation has two complex conjugate roots, meaning no real solutions exist.

In our example, we calculated the discriminant as \( -4 \), which is negative, indicating complex solutions.

Solving Quadratics

Solving quadratic equations involves finding the values of \(x\) that satisfy \( ax^2 + bx + c = 0 \). There are several methods to solve quadratics:
1. Factoring, which works well when the equation can be expressed as a product of binomials.
2. Using the quadratic formula, which is a universal method applicable to any quadratic equation.
3. Completing the square, useful for understanding the derivation of the quadratic formula and solving equations with particular forms.
In our exercise, using the quadratic formula was most appropriate due to the complex nature of the solutions.