Problem 33 - College Algebra with Corequisite Support

Question

For the following exercises, determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve. $$ x^{2}+4 x+7=0 $$

Step-by-Step Solution

Answer

The discriminant is \(-12\), indicating two complex solutions.

Step 1: Identify the Components of the Quadratic Equation

The given quadratic equation is in the form \( ax^2 + bx + c = 0 \). By comparing this with the given equation \( x^2 + 4x + 7 = 0 \), we identify the coefficients: \( a = 1 \), \( b = 4 \), and \( c = 7 \).

Step 2: Recall the Formula for the Discriminant

The formula for the discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is \( \Delta = b^2 - 4ac \). The discriminant helps determine the number and type of solutions of the quadratic equation.

Step 3: Calculate the Discriminant

Plug in the values of \( a \), \( b \), and \( c \) into the discriminant formula: \[ \Delta = 4^2 - 4 \cdot 1 \cdot 7 = 16 - 28 = -12 \]. So, the discriminant \( \Delta = -12 \).

Step 4: Analyze the Discriminant

Since the discriminant \( \Delta = -12 \) is negative, it indicates that the quadratic equation has two complex conjugate solutions. This means there are no real solutions.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of the form \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). The graph of a quadratic equation is a parabola, which can open upwards or downwards depending on the sign of \( a \). When looking at a quadratic equation, the goal is often to find the values of \( x \) where the equation equals zero, called its roots or solutions.
These solutions can be obtained through various methods, such as factoring, completing the square, or using the quadratic formula. The quadratic formula is particularly useful as it can always find the roots, no matter the nature of the solutions. The formula is:

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

In this equation, \( b^2 - 4ac \) is called the discriminant. It plays a crucial role in determining the nature of the solutions.

Complex Solutions

When discussing the solutions of a quadratic equation, the term "complex solutions" refers to solutions that are not real numbers. This happens when the quadratic equation's discriminant is negative. The discriminant, \( \Delta = b^2 - 4ac \), determines whether the solutions will be real or complex.
If the discriminant is negative, as in the case of our example \( x^2 + 4x + 7 = 0 \), we cannot find real solutions. Instead, we have complex solutions. These solutions come in pairs called complex conjugates. They are of the form:

• \( x = \frac{-b + i\sqrt{|\Delta|}}{2a} \)
• \( x = \frac{-b - i\sqrt{|\Delta|}}{2a} \)

Here, \( i \) is the imaginary unit, and \( \sqrt{|\Delta|} \) is the square root of the absolute value of the discriminant. Complex solutions are crucial for understanding problems where the scenario is not represented by real numbers alone.

Nature of Solutions

The nature of the solutions of a quadratic equation depends significantly on the discriminant \( \Delta = b^2 - 4ac \). It tells us not only the number of solutions but also their type.
Here is what the discriminant indicates about the solutions:

• If \( \Delta > 0 \), there are two distinct real solutions. The parabola will intersect the x-axis at two points.
• If \( \Delta = 0 \), there is exactly one real solution (also called a repeated or double root). The parabola will just touch the x-axis.
• If \( \Delta < 0 \), as in our example where \( \Delta = -12 \), there are no real solutions. Instead, there are two complex conjugate solutions.

Understanding the nature of solutions aids in predicting the behavior of the quadratic equation in different contexts, which is extremely useful in various fields of science and engineering.