Problem 67 - College Algebra with Modeling and Visualization

Question

Solve. Write answers in standard form. $$ x^{2}+2 x+4=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = -1 + \sqrt{3}i \) and \( x = -1 - \sqrt{3}i \).

Step 1: Identify the Standard Form

The given quadratic equation is already in standard form: \[ ax^2 + bx + c = 0 \]where \( a = 1 \), \( b = 2 \), and \( c = 4 \).

Step 2: Use the Quadratic Formula

To solve for \( x \), we can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Given \( a = 1 \), \( b = 2 \), \( c = 4 \), substitute these values into the formula.

Step 3: Compute the Discriminant

Calculate the discriminant, \( b^2 - 4ac \): \[ b^2 - 4ac = 2^2 - 4 \times 1 \times 4 = 4 - 16 = -12 \]Since the discriminant is negative, the solutions will be complex.

Step 4: Substitute into the Quadratic Formula

Now substitute \( b = 2 \), \( a = 1 \), and the discriminant into the quadratic formula:\[ x = \frac{-2 \pm \sqrt{-12}}{2} \]

Step 5: Simplify the Expression

Since \( \sqrt{-12} = \sqrt{12}i = 2\sqrt{3}i \), simplify the expression:\[ x = \frac{-2 \pm 2\sqrt{3}i}{2} \]This can be further simplified to:\[ x = -1 \pm \sqrt{3}i \]

Step 6: Write the Solution in Standard Form

The solutions of the equation are complex numbers. Therefore, the standard form for the roots \( x \) are: \[ x = -1 + \sqrt{3}i \] and \[ x = -1 - \sqrt{3}i \]

Key Concepts

Quadratic Formula

The Quadratic Formula is a powerful tool for solving any quadratic equation. A quadratic equation is typically expressed in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. This formula allows you to find the solutions of the equation, which are the values of \( x \) that make the equation true. The Quadratic Formula is given by:

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

This formula works for any quadratic equation, even when the coefficients or the solutions are complex numbers. It’s a universal solution that can provide two potential solutions (hence the plus-minus symbol). Immediate application by identifying the coefficients \( a \), \( b \), and \( c \) allows for fast resolution of quadratic equations, as seen in the example equation \( x^2 + 2x + 4 = 0 \). Here, by plugging in the values \( a = 1 \), \( b = 2 \), and \( c = 4 \), you access the roots directly.

Complex Numbers

Complex numbers may seem intimidating at first, but they follow straightforward rules. A complex number has a real part and an imaginary part, and is typically written in the form \( a + bi \), where \( i \) is the imaginary unit. The imaginary unit \( i \) is defined as \( \sqrt{-1} \), which signifies numbers that cannot be found on the traditional number line.

• The real part is the \( a \), just like any regular number you already know.
• The imaginary part consists of the \( bi \), where \( b \) is an actual number, and \( i \) indicates the presence of an imaginary component.

When faced with a quadratic equation whose discriminant is negative, like our example with the discriminant of \(-12\), the solutions will include imaginary numbers. For instance, when simplifying \( \sqrt{-12} \), you find \( \sqrt{12}i \), leading to solutions expressed as \( -1 \pm \sqrt{3}i \). Thus, understanding complex numbers enriches your ability to handle a broader range of equations.

Discriminant in Algebra

The discriminant is a key concept in understanding the nature of solutions of a quadratic equation. Calculated as \( b^2 - 4ac \), the discriminant tells us whether the roots of the equation are real or complex, and how many roots exist.

• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, there is exactly one real root (a repeated root).
• If negative, the roots are complex conjugates and are not real numbers.

In our example \( x^2 + 2x + 4 = 0 \), calculating the discriminant gives us \(-12\). This negative value indicates the solutions will include complex numbers, showcasing an essential aspect of algebra. Recognizing this pattern empowers you to predict and classify the types of solutions possible, significantly enhancing your problem-solving skills in distinguishing different equation outcomes.