Problem 4 - College Algebra and Calculus: An Applied Approach

Question

Use the discriminant to determine the number of real solutions of the quadratic equation. \(x^{2}+2 x+4=0\)

Step-by-Step Solution

Answer

The given quadratic equation \(x^{2}+2 x+4=0\) has no real solutions as the discriminant is less than zero.

Step 1: Identify the coefficients

For the quadratic equation \(x^{2}+2 x+4=0\), the coefficients are \(a=1\), \(b=2\), and \(c=4\).

Step 2: Use the Discriminant Formula

Input the identified values into the discriminant formula \(D = b^2 - 4ac\). This gives us \(D = 2^2 - 4*1*4 = 4 - 16 = -12\)

Step 3: Analyze the Discriminant

We find that \(D < 0\). So, this indicates that there are no real roots for the given quadratic equation. The roots are complex or imaginary in this case.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of the form \( ax^2 + bx + c = 0 \). It is defined by three coefficients: \( a \), \( b \), and \( c \), where \( a eq 0 \). These coefficients determine the shape and position of the parabola represented by the graph of the quadratic equation. The power of the leading term (which is 2) indicates that the graph will be a U-shaped curve called a parabola.
Understanding the structure is key:

• \( a \): The coefficient of \( x^2 \). It determines if the parabola opens upwards (\( a > 0 \)) or downwards (\( a < 0 \)).
• \( b \): The coefficient of \( x \). This affects the position of the vertex but does not affect the direction the parabola opens.
• \( c \): The constant term. It influences the y-intercept, or where the parabola crosses the y-axis.

These attributes help not only in graphing the equation but also in solving it using different methods like factoring, completing the square, or the quadratic formula.

Complex Roots

When solving a quadratic equation, the nature of the roots (solutions) depends on the discriminant, which is part of the quadratic formula. The discriminant is calculated as \( D = b^2 - 4ac \).
Here’s what the discriminant tells us about the roots:

• If \( D > 0 \), there are two distinct real roots.
• If \( D = 0 \), there is one real root (or a repeated real root).
• If \( D < 0 \), there are no real roots; the roots are complex.

Complex roots occur in conjugate pairs when the discriminant is negative. This means that they can be written in the form \( a + bi \) and \( a - bi \), where \( i \) is the imaginary unit \( \sqrt{-1} \).
For the equation \( x^2 + 2x + 4 = 0 \), the calculated discriminant \( D = -12 \) leads to complex roots, since \( -12 < 0 \). Complex roots are essential in understanding systems that cannot be analyzed with real numbers alone, especially in fields like engineering and physics.

Real Solutions

Real solutions to a quadratic equation are the values of \( x \) where the graph of the parabola intersects the x-axis. They are the real-number answers to the equation \( ax^2 + bx + c = 0 \). The concept of real solutions is crucial in practical applications where only visible and measurable outcomes are required.
To determine the real solutions, look at the discriminant:

• When \( D > 0 \), the equation has two different real solutions. The parabola crosses the x-axis at two points.
• When \( D = 0 \), there is exactly one real solution. The vertex of the parabola touches the x-axis at only one point.
• When \( D < 0 \), no real solutions exist, as the graph does not intersect the x-axis at all.

For our problem, since \( D = -12 \), there are no real solutions; the parabola does not touch or cross the x-axis. Instead, it lies entirely above or below the axis, showing that the solutions are complex, not real. Recognizing real solutions helps predict behaviors in various scenarios, from physics to finance.