Problem 14 - Algebra for College Students

Question

Solve each quadratic equation using the method that seems most appropriate to you. $$3 x^{2}-2 x+8=0$$

Step-by-Step Solution

Answer

The roots are \( x = \frac{1 \pm i\sqrt{23}}{3} \).

Step 1: Identify the Form and Set Up the Equation

The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \) where \( a = 3 \), \( b = -2 \), and \( c = 8 \). We'll first check if it can be easily factored, but since it includes complex numbers, we'll choose to use the quadratic formula.

Step 2: Recall the Quadratic Formula

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). We will use this formula to find the roots of the quadratic equation.

Step 3: Calculate the Discriminant

Calculate the discriminant \( b^2 - 4ac \). Substitute the values: \( (-2)^2 - 4 \cdot 3 \cdot 8 = 4 - 96 = -92 \). The discriminant is negative, indicating that the solutions are complex numbers.

Step 4: Substitute into the Quadratic Formula

Substitute \( a = 3 \), \( b = -2 \), and \( c = 8 \) into the quadratic formula: \[ x = \frac{-(-2) \pm \sqrt{-92}}{2 \cdot 3} = \frac{2 \pm \sqrt{-92}}{6} \].

Step 5: Simplify the Expression

Since the discriminant \( -92 \) is negative, write \( \sqrt{-92} = i \cdot \sqrt{92} \), where \( i \) is the imaginary unit. Simplify to get \( x = \frac{2 \pm i \cdot \sqrt{92}}{6} \). Further simplify \( \sqrt{92} \): \( \sqrt{92} = \sqrt{4 \cdot 23} = 2 \cdot \sqrt{23} \). Substitute back: \[ x = \frac{2 \pm 2i\sqrt{23}}{6} = \frac{1 \pm i\sqrt{23}}{3} \].

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). It allows us to find the roots of these equations, even when they can't be factored easily. The quadratic formula itself is given by:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Here, \( a \), \( b \), and \( c \) are coefficients of the quadratic equation, where \( a eq 0 \). The symbol \( \pm \) means that you'll always take both the positive and negative roots when solving. Using the quadratic formula is often the most straightforward method when dealing with complex roots, as it automatically adjusts to any discriminant value, including negative ones that indicate complex solutions. The quadratic formula is especially useful because it works for all types of quadratic equations, regardless of whether the roots are real or complex. By plugging in your values for \( a \), \( b \), and \( c \), you can solve any quadratic equation.

Complex Numbers

Complex numbers are an extension of real numbers and are especially useful in solving equations like quadratics with negative discriminants. A complex number is expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with the property \( i^2 = -1 \).

• The real part is \( a \).
• The imaginary part is \( bi \).

In the context of solving quadratic equations, when the discriminant (\( b^2 - 4ac \)) is negative, the roots of the equation are complex. This requires the use of the quadratic formula, as it accommodates negative values under the square root by introducing the imaginary unit \( i \). For example, to find the square root of \(-92\), we express it as \( \sqrt{-92} = i \cdot \sqrt{92} \). This indicates that the roots of the equation are complex, incorporating both real and imaginary parts.

Discriminant

The discriminant is a key concept in determining the nature of the roots of a quadratic equation. It is represented by the expression:

• \( b^2 - 4ac \)

The value of the discriminant provides essential information about the quadratic equation:

• If the discriminant is positive, the equation has two distinct real roots.
• If the discriminant is zero, the equation has exactly one real root (a repeated root).
• If the discriminant is negative, the equation has two complex roots.

In our exercise, the discriminant of the equation \( 3x^2 - 2x + 8 = 0 \) is calculated as \(-92\), which means the roots are complex. Understanding the discriminant helps us anticipate the nature of the roots before solving the equation itself.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is the basis for identifying and solving quadratics. It is written as:

• \( ax^2 + bx + c = 0 \)

where \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). This form is essential because it standardizes the way we recognize, analyze, and solve these types of equations.

• \( a \) is the coefficient of \( x^2 \).
• \( b \) is the coefficient of \( x \).
• \( c \) is the constant term.

Understanding the standard form allows us to easily identify the quadratic equation in any context and apply techniques such as factoring, completing the square, or using the quadratic formula. In solving the equation \( 3x^2 - 2x + 8 = 0 \), we started by recognizing its standard form, which guided the subsequent actions and calculations.