Problem 93 - College Algebra with Modeling and Visualization

Question

Complete the following. (a) Write the equation as \(a x^{2}+b x+c=0\) with \(a>0\) (b) Calculate the discriminant \(b^{2}-4 a c\) and determine the number of real solutions. (c) Solve the equation. $$ x\left(\frac{1}{2} x+1\right)=-\frac{13}{2} $$

Step-by-Step Solution

Answer

No real solutions; two complex solutions: \( x = -1 \pm 2\sqrt{3}i \).

Step 1: Expand the Equation

Start with the given equation \[ x \left( \frac{1}{2} x + 1 \right) = -\frac{13}{2} \] Distribute \( x \) to the terms inside the parenthesis:\[ \frac{1}{2} x^2 + x = -\frac{13}{2} \]

Step 2: Bring to Standard Form

Rearrange the equation to obtain a standard form of a quadratic equation:\[ \frac{1}{2} x^2 + x + \frac{13}{2} = 0 \] Multiply the entire equation by 2 to eliminate the fraction:\[ x^2 + 2x + 13 = 0 \] Here, \( a = 1 \), \( b = 2 \), and \( c = 13 \).

Step 3: Calculate the Discriminant

The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula \( b^2 - 4ac \). Substitute the values for \( a \), \( b \), and \( c \):\[ b^2 - 4ac = 2^2 - 4(1)(13) = 4 - 52 = -48 \].

Step 4: Determine the Number of Real Solutions

Since the discriminant is \(-48\), which is less than zero, the quadratic equation has no real solutions. It has two complex solutions instead.

Step 5: Solve the Equation Using Complex Numbers

Use the quadratic formula to find the complex solutions:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-2 \pm \sqrt{-48}}{2} \]Simplify:\[ x = \frac{-2 \pm \sqrt{48}i}{2} \]\[ x = \frac{-2 \pm 4\sqrt{3}i}{2} \]Divide each term by 2:\[ x = -1 \pm 2\sqrt{3}i \]The solutions are \( x = -1 + 2\sqrt{3}i \) and \( x = -1 - 2\sqrt{3}i \).

Key Concepts

Discriminant

The discriminant is a crucial part of understanding quadratic equations. It is represented by the expression \( b^2 - 4ac \), where \( a \), \( b \), and \( c \) are coefficients from the quadratic equation \( ax^2 + bx + c = 0 \). The discriminant helps us determine the nature of the roots of the quadratic equation.

• If the discriminant is greater than zero, the quadratic equation has two distinct real roots.
• If the discriminant is exactly zero, there is exactly one real root, or in other words, a repeated root.
• When the discriminant is less than zero, as in this exercise, the quadratic equation does not have real solutions and instead has two complex conjugate roots.

By calculating the discriminant, you can quickly assess what types of roots to expect without solving the entire equation. In this specific exercise, the calculated discriminant was \(-48\), confirming the existence of complex roots.

Complex Numbers

Complex numbers are an extension of the real numbers that enable solutions to equations that have no real solutions. They can be expressed in the form \( a + bi \), where \( i \) is the imaginary unit, with the property that \( i^2 = -1 \). This imaginary unit allows us to handle the square roots of negative numbers, which are not defined within the real numbers.

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

In the given exercise, when the discriminant was less than zero, we discovered complex solutions. The roots \( x = -1 \pm 2\sqrt{3}i \) consist of a real part \(-1\), and an imaginary part \( 2\sqrt{3}i \). Understanding complex numbers is essential to handling cases where real solutions are impossible, providing a visually deeper insight into the behavior of quadratic equations.

Quadratic Formula

The quadratic formula is a universal tool used to solve quadratic equations of the form \( ax^2 + bx + c = 0 \). It is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula provides the roots of the quadratic equation by incorporating the coefficients and the discriminant. It works in any scenario, whether the equation has real or complex solutions. Here is how it functions:

• The term \( -b \pm \sqrt{b^2 - 4ac} \) illustrates how roots can diverge based on the discriminant's value.
• The division by \( 2a \) scales the effect of these solutions based on the leading coefficient of the equation.

In the original problem, the quadratic formula allowed calculation of the complex solutions \( x = -1 + 2\sqrt{3}i \) and \( x = -1 - 2\sqrt{3}i \). Mastery of this formula enhances the ability to solve not only equations with real roots but also those involving complex numbers. Its reliability stems from producing consistent solutions across all types of quadratic equations.