Problem 28 - Algebra A Combined Function

Question

Use the quadratic formula to solve each equation. These equations have real solutions and complex, but not real, solutions. $$ 9 x^{2}+x+2=0 $$

Step-by-Step Solution

Answer

The equation \( 9x^2 + x + 2 = 0 \) has complex solutions \( x = \frac{-1}{18} \pm \frac{\sqrt{71}i}{18} \).

Step 1: Identify a, b, and c

First, write down the standard form of a quadratic equation which is \( ax^2 + bx + c = 0 \). For the equation \( 9x^2 + x + 2 = 0 \), identify the coefficients as \( a = 9 \), \( b = 1 \), and \( c = 2 \).

Step 2: Write the Quadratic Formula

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

Step 3: Calculate the Discriminant

The discriminant \( \Delta \) is given by \( \Delta = b^2 - 4ac \). Plug in the values: \( \Delta = 1^2 - 4(9)(2) = 1 - 72 = -71 \). Since \( \Delta < 0 \), this equation has two complex solutions.

Step 4: Compute the Real and Imaginary Parts

Since the discriminant is negative, calculate the real part and the imaginary part separately. Using the quadratic formula, \( x = \frac{-1 \pm \sqrt{-71}}{18} \). This can be written as \( x = \frac{-1}{18} \pm \frac{\sqrt{71}i}{18} \), where \( i \) is the imaginary unit.

Key Concepts

Understanding the Discriminant

The discriminant is a crucial part of solving quadratic equations, particularly when using the quadratic formula. It's found within the quadratic formula: \[ \Delta = b^2 - 4ac \] This part of the formula tells us a lot about the nature of the solutions to the quadratic equation.

• If \( \Delta > 0 \), the equation has two distinct real solutions.
• If \( \Delta = 0 \), the equation has exactly one real solution, or we can call it a repeated solution.
• If \( \Delta < 0 \), the equation doesn't have real solutions but instead has two complex solutions.

For the given equation, the discriminant is calculated as\[ 1^2 - 4(9)(2) = 1 - 72 = -71 \]This negative result tells us that the solutions to our equation are complex, not real.

Complex Solutions in Quadratic Equations

Complex solutions arise in quadratic equations when the discriminant \( \Delta \) is less than zero. This occurs because the square root of a negative number isn't defined within the real numbers.Here's where complex numbers come in, centered around the imaginary unit \( i \), defined as \( i^2 = -1 \). For the equation \( 9x^2 + x + 2 = 0 \), the solutions are computed by finding the roots of\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-1 \pm \sqrt{-71}}{18} \]This can be expressed with a real part and an imaginary part:

• Real part: \( \frac{-1}{18} \)
• Imaginary part: \( \pm \frac{\sqrt{71}i}{18} \)

Therefore, the complex solutions for \( x \) are \( \frac{-1}{18} + \frac{\sqrt{71}i}{18} \) and \( \frac{-1}{18} - \frac{\sqrt{71}i}{18} \). These solutions are not visible on the real number line, but they are essential in understanding the full scope of a quadratic's behavior.

The Nature of Quadratic Equations

Quadratic equations appear in the form:\[ ax^2 + bx + c = 0 \]where \( a \), \( b \), and \( c \) are coefficients, and \( a eq 0 \). Solving these equations can be done using various methods, such as factoring, completing the square, or using the quadratic formula.The quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]is often a reliable choice as it can find solutions for any quadratic equation, whether they are real or complex. The discriminant \( b^2 - 4ac \) within this formula helps assess the type of solutions expected:

• Two distinct real solutions when \( \Delta > 0 \)
• One real, repeated solution when \( \Delta = 0 \)
• Two complex solutions when \( \Delta < 0 \)

In problems involving quadratic equations, recognizing these solution types helps apply the appropriate method to solve them and to understand the nature of the roots better.